What is the general solution for cos(45+x)-cos(45-x)=sqrt(2)cos(x) ?

The general solution for cos(45+x)-cos(45-x)=sqrt(2)cos(x) is x=135+180n

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cos(45+x)-cos(45-x)=sqrt(2)cos(x)

Solution

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

Solution

x = 13 5^{\circ} + 18 0^{\circ} n

Radians

x = \frac{3 π}{4} + πn

Solution steps

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

Rewrite using trig identities

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

Rewrite using trig identities

cos (4 5^{\circ} - x)

Use the Angle Difference identity:

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

= cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

Simplify

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x) : \frac{2 cos ( x ) + 2 sin ( x )}{2}

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

cos (4 5^{\circ}) cos (x) = \frac{2 cos ( x )}{2}

cos (4 5^{\circ}) cos (x)

Simplify

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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)

Multiply fractions:

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

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

sin (4 5^{\circ}) sin (x) = \frac{2 sin ( x )}{2}

sin (4 5^{\circ}) sin (x)

Simplify

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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} sin (x)

Multiply fractions:

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

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

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

Use the Angle Sum identity:

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

= cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

Simplify

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x) : \frac{2 cos ( x ) - 2 sin ( x )}{2}

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

cos (4 5^{\circ}) cos (x) = \frac{2 cos ( x )}{2}

cos (4 5^{\circ}) cos (x)

Simplify

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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)

Multiply fractions:

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

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

sin (4 5^{\circ}) sin (x) = \frac{2 sin ( x )}{2}

sin (4 5^{\circ}) sin (x)

Simplify

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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} sin (x)

Multiply fractions:

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

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

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

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

Simplify

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

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

Expand

2 cos (x) - 2 sin (x) - (2 cos (x) + 2 sin (x)) : - 22 sin (x)

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

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

Simplify

2 cos (x) - 2 sin (x) - 2 cos (x) - 2 sin (x) : - 22 sin (x)

2 cos (x) - 2 sin (x) - 2 cos (x) - 2 sin (x)

Add similar elements:

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

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

Add similar elements:

- 2 sin (x) - 2 sin (x) = - 22 sin (x)

= - 22 sin (x)

= - 22 sin (x)

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

= - 2 sin (x)

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

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

Subtract

2 cos (x)

from both sides

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

- 2 tan (x) - 2 = 0

- 2 tan (x) - 2 = 0

Move

2

to the right side

- 2 tan (x) - 2 = 0

Add

2

to both sides

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

Simplify

- 2 tan (x) = 2

- 2 tan (x) = 2

Divide both sides by

- 2

- 2 tan (x) = 2

Divide both sides by

- 2

\frac{- 2 tan ( x )}{- 2} = \frac{2}{- 2}

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 13 5^{\circ} + 18 0^{\circ} n

x = 13 5^{\circ} + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(45+x)-cos(45-x)=sqrt(2)cos(x) ?
The general solution for cos(45+x)-cos(45-x)=sqrt(2)cos(x) is x=135+180n