Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

Apply the sum of angles identity.

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Rewrite using the commutative property of multiplication.

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

Add $2 \sin (x) \cos (y)$ and $0$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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Step 6.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.3.2

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

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.2.6

Add $2 \cos (x) \cos (y)$ and $0$ .

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

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 6.4

Cancel the common factor of $\cos (y)$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)}$

Step 6.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x)$

Step 7

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

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