What is the general solution for tan(pi/4-x)+tan(x)-1=0 ?

The general solution for tan(pi/4-x)+tan(x)-1=0 is x=2pin,x=pi+2pin,x= pi/4+pin

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Popular >

tan(pi/4-x)+tan(x)-1=0

Solution

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Rewrite using trig identities

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Rewrite using trig identities

tan (\frac{π}{4} - x)

Use the basic trigonometric identity:

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

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

Use the Angle Difference identity:

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

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

Use the Angle Difference identity:

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

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

Simplify

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

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

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

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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}

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

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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}

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

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

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

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

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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}

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

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

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

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

Simplify

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1 : \frac{- 2 sin ( x ) + tan ( x ) cos ( x ) + tan ( x ) sin ( x )}{cos ( x ) + sin ( x )}

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1

Convert element to fraction:

tan (x) = \frac{tan ( x ) ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}, 1 = \frac{1 ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}

= \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + \frac{tan ( x ) ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )} - \frac{1 \cdot ( cos ( x ) + sin ( 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 ) + tan ( x ) ( cos ( x ) + sin ( x ) ) - 1 \cdot ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}

Multiply:

1 \cdot (cos (x) + sin (x)) = (cos (x) + sin (x))

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

Expand

cos (x) - sin (x) + tan (x) (cos (x) + sin (x)) - (cos (x) + sin (x)) : - 2 sin (x) + tan (x) cos (x) + tan (x) sin (x)

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

Expand

tan (x) (cos (x) + sin (x)) : tan (x) cos (x) + tan (x) sin (x)

tan (x) (cos (x) + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

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

= tan (x) cos (x) + tan (x) sin (x)

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - cos (x) - sin (x)

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

Simplify

cos (x) - sin (x) + tan (x) cos (x) + tan (x) sin (x) - cos (x) - sin (x) : - 2 sin (x) + tan (x) cos (x) + tan (x) sin (x)

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

Add similar elements:

cos (x) - cos (x) = 0

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

Add similar elements:

- sin (x) - sin (x) = - 2 sin (x)

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

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

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

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

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

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

Rewrite using trig identities

- 2 sin (x) + cos (x) tan (x) + sin (x) tan (x)

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x)

= - 2 sin (x) + sin (x) + sin (x) tan (x)

Simplify

= - sin (x) + sin (x) tan (x)

- sin (x) + sin (x) tan (x) = 0

Factor

- sin (x) + sin (x) tan (x) : sin (x) (tan (x) - 1)

- sin (x) + sin (x) tan (x)

Factor out common term

sin (x)

= sin (x) (- 1 + tan (x))

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

Solving each part separately

sin (x) = 0 or tan (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(pi/4-x)+tan(x)-1=0 ?
The general solution for tan(pi/4-x)+tan(x)-1=0 is x=2pin,x=pi+2pin,x= pi/4+pin