Trigonometry Examples

$\cos (x - y) - \cos (x + y) = 2 \sin (x) \sin (y)$

Step 1

Start on the left side.

$\cos (x - y) - \cos (x + y)$

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\cos (x) \cos (- y) - \sin (x) \sin (- y) - \cos (x + y)$

Step 3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\cos (x) \cos (- y) - \sin (x) \sin (- y) - (\cos (x) \cos (y) - \sin (x) \sin (y))$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Since $\cos (- y)$ is an even function, rewrite $\cos (- y)$ as $\cos (y)$ .

$\cos (x) \cos (y) - \sin (x) \sin (- y) - (\cos (x) \cos (y) - \sin (x) \sin (y))$

Step 4.1.2

Since $\sin (- y)$ is an odd function, rewrite $\sin (- y)$ as $- \sin (y)$ .

$\cos (x) \cos (y) - \sin (x) (- \sin (y)) - (\cos (x) \cos (y) - \sin (x) \sin (y))$

Step 4.1.3

Multiply $- \sin (x) (- \sin (y))$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$\cos (x) \cos (y) + 1 \sin (x) \sin (y) - (\cos (x) \cos (y) - \sin (x) \sin (y))$

Step 4.1.3.2

Multiply $\sin (x)$ by $1$ .

$\cos (x) \cos (y) + \sin (x) \sin (y) - (\cos (x) \cos (y) - \sin (x) \sin (y))$

Step 4.1.4

Apply the distributive property.

$\cos (x) \cos (y) + \sin (x) \sin (y) - (\cos (x) \cos (y)) - (- \sin (x) \sin (y))$

Step 4.1.5

Multiply $- (- \sin (x) \sin (y))$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

$\cos (x) \cos (y) + \sin (x) \sin (y) - \cos (x) \cos (y) + 1 (\sin (x) \sin (y))$

Step 4.1.5.2

Multiply $\sin (x)$ by $1$ .

$\cos (x) \cos (y) + \sin (x) \sin (y) - \cos (x) \cos (y) + \sin (x) \sin (y)$

Step 4.2

Combine the opposite terms in $\cos (x) \cos (y) + \sin (x) \sin (y) - \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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Step 4.2.1

Subtract $\cos (x) \cos (y)$ from $\cos (x) \cos (y)$ .

$\sin (x) \sin (y) + 0 + \sin (x) \sin (y)$

Step 4.2.2

Add $\sin (x) \sin (y)$ and $0$ .

$\sin (x) \sin (y) + \sin (x) \sin (y)$

Step 4.3

Add $\sin (x) \sin (y)$ and $\sin (x) \sin (y)$ .

$2 \sin (x) \sin (y)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos (x - y) - \cos (x + y) = 2 \sin (x) \sin (y)$ is an identity