Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

Reorder terms.

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

Step 4

Now consider the right side of the equation.

$\tan (x) + \tan (y)$

Step 5

Convert to sines and cosines.

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Step 5.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 5.2

Write $\tan (y)$ in sines and cosines using the quotient identity.

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

Step 6

Add fractions.

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Step 6.1

To write $\frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (y)}{\cos (y)}$ .

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

Step 6.2

To write $\frac{\sin (y)}{\cos (y)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 6.3

Write each expression with a common denominator of $\cos (x) \cos (y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\cos (y)}{\cos (y)}$ .

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

Step 6.3.2

Multiply $\frac{\sin (y)}{\cos (y)}$ by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 6.3.3

Reorder the factors of $\cos (y) \cos (x)$ .

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 7

Reorder terms.

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

Step 8

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

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