Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

Apply the sum of angles identity.

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

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 $\cos (y)$ .

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

Step 4.1.2

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

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

Step 4.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.2

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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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 5

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