What is the general solution for tan((5pi)/4+x)+tan((5pi)/4-x)=4 ?

The general solution for tan((5pi)/4+x)+tan((5pi)/4-x)=4 is x=(5pi)/6+pin,x= pi/6+pin

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Popular Trigonometry >

tan((5pi)/4+x)+tan((5pi)/4-x)=4

Solution

tan (\frac{5 π}{4} + x) + tan (\frac{5 π}{4} - x) = 4

Solution

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Degrees

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

tan (\frac{5 π}{4} + x) + tan (\frac{5 π}{4} - x) = 4

Rewrite using trig identities

tan (\frac{5 π}{4} + x) + tan (\frac{5 π}{4} - x) = 4

Rewrite using trig identities

tan (\frac{5 π}{4} + x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( \frac{5 π}{4} + x )}{cos ( \frac{5 π}{4} + x )}

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( \frac{5 π}{4} ) cos ( x ) + cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} + x )}

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( \frac{5 π}{4} ) cos ( x ) + cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) - sin ( \frac{5 π}{4} ) sin ( x )}

Simplify

\frac{sin ( \frac{5 π}{4} ) cos ( x ) + cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) - sin ( \frac{5 π}{4} ) sin ( x )} : - \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )}

\frac{sin ( \frac{5 π}{4} ) cos ( x ) + cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) - sin ( \frac{5 π}{4} ) sin ( x )}

sin (\frac{5 π}{4}) cos (x) + cos (\frac{5 π}{4}) sin (x) = - \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

sin (\frac{5 π}{4}) cos (x) + cos (\frac{5 π}{4}) sin (x)

sin (\frac{5 π}{4}) = - \frac{2}{2}

sin (\frac{5 π}{4})

Rewrite using trig identities:

sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

sin (\frac{5 π}{4})

Write

sin (\frac{5 π}{4})

sin (π + \frac{π}{4})

= sin (π + \frac{π}{4})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 0 \cdot \frac{2}{2} + (- 1) \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) + cos (\frac{5 π}{4}) sin (x)

cos (\frac{5 π}{4}) = - \frac{2}{2}

cos (\frac{5 π}{4})

Rewrite using trig identities:

cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

cos (\frac{5 π}{4})

Write

cos (\frac{5 π}{4})

cos (π + \frac{π}{4})

= cos (π + \frac{π}{4})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= (- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

= \frac{- \frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) - sin ( \frac{5 π}{4} ) sin ( x )}

cos (\frac{5 π}{4}) cos (x) - sin (\frac{5 π}{4}) sin (x) = - \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (\frac{5 π}{4}) cos (x) - sin (\frac{5 π}{4}) sin (x)

cos (\frac{5 π}{4}) = - \frac{2}{2}

cos (\frac{5 π}{4})

Rewrite using trig identities:

cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

cos (\frac{5 π}{4})

Write

cos (\frac{5 π}{4})

cos (π + \frac{π}{4})

= cos (π + \frac{π}{4})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= (- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - sin (\frac{5 π}{4}) sin (x)

sin (\frac{5 π}{4}) = - \frac{2}{2}

sin (\frac{5 π}{4})

Rewrite using trig identities:

sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

sin (\frac{5 π}{4})

Write

sin (\frac{5 π}{4})

sin (π + \frac{π}{4})

= sin (π + \frac{π}{4})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 0 \cdot \frac{2}{2} + (- 1) \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - (- \frac{2}{2} sin (x))

Apply rule

- (- a) = a

= - \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

= \frac{- \frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{- \frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}

Multiply

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{- \frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}

Multiply

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{- \frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiply

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiply

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{- \frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Combine the fractions

- \frac{2 cos ( x )}{2} + \frac{2 sin ( x )}{2} : \frac{- 2 cos ( x ) + 2 sin ( x )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ( x ) + 2 sin ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{- 2 c o s ( x ) + 2 s i n ( x )}{2}}

Combine the fractions

- \frac{2 cos ( x )}{2} - \frac{2 sin ( x )}{2} : \frac{- 2 cos ( x ) - 2 sin ( x )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ( x ) - 2 sin ( x )}{2}

= \frac{\frac{- 2 c o s ( x ) - 2 s i n ( x )}{2}}{\frac{- 2 c o s ( x ) + 2 s i n ( x )}{2}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( - 2 cos ( x ) - 2 sin ( x ) ) \cdot 2}{2 ( - 2 cos ( x ) + 2 sin ( x ) )}

Cancel the common factor:

2

= \frac{- 2 cos ( x ) - 2 sin ( x )}{- 2 cos ( x ) + 2 sin ( x )}

Factor out common term

2

= - \frac{2 ( cos ( x ) + sin ( x ) )}{- 2 cos ( x ) + 2 sin ( x )}

Factor out common term

2

= - \frac{2 ( cos ( x ) + sin ( x ) )}{2 ( - cos ( x ) + sin ( x ) )}

Cancel the common factor:

2

= - \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )}

= - \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )}

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( \frac{5 π}{4} - x )}{cos ( \frac{5 π}{4} - x )}

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( \frac{5 π}{4} ) cos ( x ) - cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} - x )}

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{sin ( \frac{5 π}{4} ) cos ( x ) - cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) + sin ( \frac{5 π}{4} ) sin ( x )}

Simplify

\frac{sin ( \frac{5 π}{4} ) cos ( x ) - cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) + sin ( \frac{5 π}{4} ) sin ( x )} : - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

\frac{sin ( \frac{5 π}{4} ) cos ( x ) - cos ( \frac{5 π}{4} ) sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) + sin ( \frac{5 π}{4} ) sin ( x )}

sin (\frac{5 π}{4}) cos (x) - cos (\frac{5 π}{4}) sin (x) = - \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

sin (\frac{5 π}{4}) cos (x) - cos (\frac{5 π}{4}) sin (x)

sin (\frac{5 π}{4}) = - \frac{2}{2}

sin (\frac{5 π}{4})

Rewrite using trig identities:

sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

sin (\frac{5 π}{4})

Write

sin (\frac{5 π}{4})

sin (π + \frac{π}{4})

= sin (π + \frac{π}{4})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 0 \cdot \frac{2}{2} + (- 1) \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - cos (\frac{5 π}{4}) sin (x)

cos (\frac{5 π}{4}) = - \frac{2}{2}

cos (\frac{5 π}{4})

Rewrite using trig identities:

cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

cos (\frac{5 π}{4})

Write

cos (\frac{5 π}{4})

cos (π + \frac{π}{4})

= cos (π + \frac{π}{4})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= (- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - (- \frac{2}{2} sin (x))

Apply rule

- (- a) = a

= - \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

= \frac{- \frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{cos ( \frac{5 π}{4} ) cos ( x ) + sin ( \frac{5 π}{4} ) sin ( x )}

cos (\frac{5 π}{4}) cos (x) + sin (\frac{5 π}{4}) sin (x) = - \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (\frac{5 π}{4}) cos (x) + sin (\frac{5 π}{4}) sin (x)

cos (\frac{5 π}{4}) = - \frac{2}{2}

cos (\frac{5 π}{4})

Rewrite using trig identities:

cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

cos (\frac{5 π}{4})

Write

cos (\frac{5 π}{4})

cos (π + \frac{π}{4})

= cos (π + \frac{π}{4})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

= cos (π) cos (\frac{π}{4}) - sin (π) sin (\frac{π}{4})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= (- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) + sin (\frac{5 π}{4}) sin (x)

sin (\frac{5 π}{4}) = - \frac{2}{2}

sin (\frac{5 π}{4})

Rewrite using trig identities:

sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

sin (\frac{5 π}{4})

Write

sin (\frac{5 π}{4})

sin (π + \frac{π}{4})

= sin (π + \frac{π}{4})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

= sin (π) cos (\frac{π}{4}) + cos (π) sin (\frac{π}{4})

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 0 \cdot \frac{2}{2} + (- 1) \frac{2}{2}

Simplify

= - \frac{2}{2}

= - \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

= \frac{- \frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{- \frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}

Multiply

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{- \frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

Multiply

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{- \frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiply

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}{- \frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiply

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{- \frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Combine the fractions

- \frac{2 cos ( x )}{2} - \frac{2 sin ( x )}{2} : \frac{- 2 cos ( x ) - 2 sin ( x )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ( x ) - 2 sin ( x )}{2}

= \frac{- \frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{- 2 c o s ( x ) - 2 s i n ( x )}{2}}

Combine the fractions

- \frac{2 cos ( x )}{2} + \frac{2 sin ( x )}{2} : \frac{- 2 cos ( x ) + 2 sin ( x )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ( x ) + 2 sin ( x )}{2}

= \frac{\frac{- 2 c o s ( x ) + 2 s i n ( x )}{2}}{\frac{- 2 c o s ( x ) - 2 s i n ( x )}{2}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( - 2 cos ( x ) + 2 sin ( x ) ) \cdot 2}{2 ( - 2 cos ( x ) - 2 sin ( x ) )}

Cancel the common factor:

2

= \frac{- 2 cos ( x ) + 2 sin ( x )}{- 2 cos ( x ) - 2 sin ( x )}

Factor out common term

2

= \frac{2 ( - cos ( x ) + sin ( x ) )}{- 2 cos ( x ) - 2 sin ( x )}

Factor out common term

2

= - \frac{2 ( - cos ( x ) + sin ( x ) )}{2 ( cos ( x ) + sin ( x ) )}

Cancel the common factor:

2

= - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

= - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

- \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )} - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} = 4

Simplify

- \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )} - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} : \frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

- \frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )} - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

Least Common Multiplier of

sin (x) - cos (x), cos (x) + sin (x) : (sin (x) - cos (x)) (cos (x) + sin (x))

sin (x) - cos (x), cos (x) + sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x) - cos (x)

cos (x) + sin (x)

= (sin (x) - cos (x)) (cos (x) + sin (x))

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

(sin (x) - cos (x)) (cos (x) + sin (x))

For

\frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )} :

multiply the denominator and numerator by

cos (x) + sin (x)

\frac{cos ( x ) + sin ( x )}{sin ( x ) - cos ( x )} = \frac{( cos ( x ) + sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} = \frac{( cos ( x ) + sin ( x ) ) ^{2}}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

For

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} :

multiply the denominator and numerator by

sin (x) - cos (x)

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} = \frac{( sin ( x ) - cos ( x ) ) ( sin ( x ) - cos ( x ) )}{( cos ( x ) + sin ( x ) ) ( sin ( x ) - cos ( x ) )} = \frac{( sin ( x ) - cos ( x ) ) ^{2}}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

= - \frac{( cos ( x ) + sin ( x ) ) ^{2}}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} - \frac{( sin ( x ) - cos ( x ) ) ^{2}}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- ( cos ( x ) + sin ( x ) ) ^{2} - ( sin ( x ) - cos ( x ) ) ^{2}}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

Expand

- (cos (x) + sin (x))^{2} - (sin (x) - cos (x))^{2} : - 2 cos^{2} (x) - 2 sin^{2} (x)

- (cos (x) + sin (x))^{2} - (sin (x) - cos (x))^{2}

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

= - (cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)) - (sin (x) - cos (x))^{2}

(sin (x) - cos (x))^{2} : sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x)

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = sin (x), b = cos (x)

= sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x)

= - (cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)) - (sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x))

- (cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)) : - cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x)

- (cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x))

Distribute parentheses

= - (cos^{2} (x)) - (2 cos (x) sin (x)) - (sin^{2} (x))

Apply minus-plus rules

+ (- a) = - a

= - cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x)

= - cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x) - (sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x))

- (sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x)) : - sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

- (sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x))

Distribute parentheses

= - (sin^{2} (x)) - (- 2 sin (x) cos (x)) - (cos^{2} (x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

= - cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x) - sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

Simplify

- cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x) - sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x) : - 2 cos^{2} (x) - 2 sin^{2} (x)

- cos^{2} (x) - 2 cos (x) sin (x) - sin^{2} (x) - sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

Add similar elements:

- 2 cos (x) sin (x) + 2 sin (x) cos (x) = 0

= - cos^{2} (x) - sin^{2} (x) - sin^{2} (x) - cos^{2} (x)

Add similar elements:

- cos^{2} (x) - cos^{2} (x) = - 2 cos^{2} (x)

= - 2 cos^{2} (x) - sin^{2} (x) - sin^{2} (x)

Add similar elements:

- sin^{2} (x) - sin^{2} (x) = - 2 sin^{2} (x)

= - 2 cos^{2} (x) - 2 sin^{2} (x)

= - 2 cos^{2} (x) - 2 sin^{2} (x)

= \frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} = 4

\frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} = 4

Subtract

4

from both sides

\frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} - 4 = 0

Simplify

\frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} - 4 : \frac{2 cos ^{2} ( x ) - 6 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} - 4

Convert element to fraction:

4 = \frac{4 ( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

= \frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} - \frac{4 ( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ^{2} ( x ) - 2 sin ^{2} ( x ) - 4 ( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

Expand

- 2 cos^{2} (x) - 2 sin^{2} (x) - 4 (sin (x) - cos (x)) (cos (x) + sin (x)) : 2 cos^{2} (x) - 6 sin^{2} (x)

- 2 cos^{2} (x) - 2 sin^{2} (x) - 4 (sin (x) - cos (x)) (cos (x) + sin (x))

Expand

- 4 (sin (x) - cos (x)) (cos (x) + sin (x)) : - 4 sin^{2} (x) + 4 cos^{2} (x)

Expand

(sin (x) - cos (x)) (cos (x) + sin (x)) : sin^{2} (x) - cos^{2} (x)

(sin (x) - cos (x)) (cos (x) + sin (x))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = sin (x), b = cos (x)

= sin^{2} (x) - cos^{2} (x)

= - 4 (sin^{2} (x) - cos^{2} (x))

Expand

- 4 (sin^{2} (x) - cos^{2} (x)) : - 4 sin^{2} (x) + 4 cos^{2} (x)

- 4 (sin^{2} (x) - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = sin^{2} (x), c = cos^{2} (x)

= - 4 sin^{2} (x) - (- 4) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 4 sin^{2} (x) + 4 cos^{2} (x)

= - 4 sin^{2} (x) + 4 cos^{2} (x)

= - 2 cos^{2} (x) - 2 sin^{2} (x) - 4 sin^{2} (x) + 4 cos^{2} (x)

Simplify

- 2 cos^{2} (x) - 2 sin^{2} (x) - 4 sin^{2} (x) + 4 cos^{2} (x) : 2 cos^{2} (x) - 6 sin^{2} (x)

- 2 cos^{2} (x) - 2 sin^{2} (x) - 4 sin^{2} (x) + 4 cos^{2} (x)

Add similar elements:

- 2 cos^{2} (x) + 4 cos^{2} (x) = 2 cos^{2} (x)

= 2 cos^{2} (x) - 2 sin^{2} (x) - 4 sin^{2} (x)

Add similar elements:

- 2 sin^{2} (x) - 4 sin^{2} (x) = - 6 sin^{2} (x)

= 2 cos^{2} (x) - 6 sin^{2} (x)

= 2 cos^{2} (x) - 6 sin^{2} (x)

= \frac{2 cos ^{2} ( x ) - 6 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{2 cos ^{2} ( x ) - 6 sin ^{2} ( x )}{( sin ( x ) - cos ( x ) ) ( cos ( x ) + sin ( x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cos^{2} (x) - 6 sin^{2} (x) = 0

Factor

2 cos^{2} (x) - 6 sin^{2} (x) : 2 (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

2 cos^{2} (x) - 6 sin^{2} (x)

Rewrite

- 6

3 \cdot 2

= 2 cos^{2} (x) + 3 \cdot 2 sin^{2} (x)

Factor out common term

2

= 2 (cos^{2} (x) - 3 sin^{2} (x))

Factor

cos^{2} (x) - 3 sin^{2} (x) : (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

cos^{2} (x) - 3 sin^{2} (x)

Rewrite

cos^{2} (x) - 3 sin^{2} (x)

cos^{2} (x) - (3 sin (x))^{2}

cos^{2} (x) - 3 sin^{2} (x)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= cos^{2} (x) - (3)^{2} sin^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} sin^{2} (x) = (3 sin (x))^{2}

= cos^{2} (x) - (3 sin (x))^{2}

= cos^{2} (x) - (3 sin (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (x) - (3 sin (x))^{2} = (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

= (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

= 2 (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

2 (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x)) = 0

Solving each part separately

cos (x) + 3 sin (x) = 0 or cos (x) - 3 sin (x) = 0

cos (x) + 3 sin (x) = 0 : x = \frac{5 π}{6} + πn

cos (x) + 3 sin (x) = 0

Rewrite using trig identities

cos (x) + 3 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) + 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 + \frac{3 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 + 3 tan (x) = 0

1 + 3 tan (x) = 0

Move

1

to the right side

1 + 3 tan (x) = 0

Subtract

1

from both sides

1 + 3 tan (x) - 1 = 0 - 1

Simplify

3 tan (x) = - 1

3 tan (x) = - 1

Divide both sides by

3

3 tan (x) = - 1

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

General solutions for

tan (x) = - \frac{3}{3}

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

cos (x) - 3 sin (x) = 0 : x = \frac{π}{6} + πn

cos (x) - 3 sin (x) = 0

Rewrite using trig identities

cos (x) - 3 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) - 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 - \frac{3 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 - 3 tan (x) = 0

1 - 3 tan (x) = 0

Move

1

to the right side

1 - 3 tan (x) = 0

Subtract

1

from both sides

1 - 3 tan (x) - 1 = 0 - 1

Simplify

- 3 tan (x) = - 1

- 3 tan (x) = - 1

Divide both sides by

- 3

- 3 tan (x) = - 1

Divide both sides by

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} : tan (x)

\frac{- 3 tan ( x )}{- 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 1}{- 3} : \frac{3}{3}

\frac{- 1}{- 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

General solutions for

tan (x) = \frac{3}{3}

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Graph

Popular Examples

-8cos(8x)=0

- 8 cos (8 x) = 0

tan^2(x)+7tan(x)+9=0

tan^{2} (x) + 7 tan (x) + 9 = 0

4cos(x)=sec(x)+3,0<= x<2pi

4 cos (x) = sec (x) + 3, 0 \leq x < 2 π

solvefor x,-1/(2y^2)=3sin(x)-1/8

so l v e f or x, - \frac{1}{2 y ^{2}} = 3 sin (x) - \frac{1}{8}

sin(θ)=(150sin(115))/(212.6)

sin (θ) = \frac{150 sin ( 11 5 ^{\circ} )}{212.6}

Frequently Asked Questions (FAQ)

What is the general solution for tan((5pi)/4+x)+tan((5pi)/4-x)=4 ?
The general solution for tan((5pi)/4+x)+tan((5pi)/4-x)=4 is x=(5pi)/6+pin,x= pi/6+pin