Algebra Examples

$\sin (x) \cos (y) = \frac{1}{2} (\sin (x + y) + \sin (x - y))$

Start on the right side.

$\frac{1}{2} (\sin (x + y) + \sin (x - y))$

Apply the sum of angles identity.

$\frac{1}{2} (\sin (x) \cos (y) + \cos (x) \sin (y) + \sin (x - y))$

Apply the sum of angles identity.

$\frac{1}{2} (\sin (x) \cos (y) + \cos (x) \sin (y) + \sin (x) \cos (- y) + \cos (x) \sin (- y))$

Simplify the expression.

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Simplify each term.

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

$\frac{1}{2} (\sin (x) \cos (y) + \cos (x) \sin (y) + \sin (x) \cos (y) + \cos (x) \sin (- y))$

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

$\frac{1}{2} (\sin (x) \cos (y) + \cos (x) \sin (y) + \sin (x) \cos (y) + \cos (x) (- \sin (y)))$

Rewrite using the commutative property of multiplication.

$\frac{1}{2} (\sin (x) \cos (y) + \cos (x) \sin (y) + \sin (x) \cos (y) - \cos (x) \sin (y))$

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

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

$\frac{1}{2} (\sin (x) \cos (y) + \sin (x) \cos (y) + 0)$

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

$\frac{1}{2} (\sin (x) \cos (y) + \sin (x) \cos (y))$

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

$\frac{1}{2} (2 \sin (x) \cos (y))$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 \sin (x) \cos (y)$ .

$\frac{1}{2} (2 (\sin (x) \cos (y)))$

Cancel the common factor.

$\frac{1}{2} (2 (\sin (x) \cos (y)))$

Rewrite the expression.

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

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

$\sin (x) \cos (y) = \frac{1}{2} \cdot (\sin (x + y) + \sin (x - y))$ is an identity