Trigonometry Examples

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

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

$- 1$ .

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

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 4.2

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

Add

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

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

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

Step 4.2.2

Add

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

$0$ .

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

Step 4.3

Add

$\cos (x) \cos (y)$ and

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

$2 \cos (x) \cos (y)$

Step 5

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

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