Precalculus Examples

$\frac{\sin (a + B)}{\cos (a) \cos (B)} = \tan (a) + \tan (B)$

Step 1

Start on the left side.

$\frac{\sin (a + B)}{\cos (a) \cos (B)}$

Step 2

Apply the sum of angles identity.

$\frac{\sin (a) \cos (B) + \cos (a) \sin (B)}{\cos (a) \cos (B)}$

Step 3

Reorder terms.

$\frac{\cos (B) \sin (a) + \cos (a) \sin (B)}{\cos (B) \cos (a)}$

Step 4

Now consider the right side of the equation.

$\tan (a) + \tan (B)$

Step 5

Convert to sines and cosines.

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Step 5.1

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

$\frac{\sin (a)}{\cos (a)} + \tan (B)$

Step 5.2

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

$\frac{\sin (a)}{\cos (a)} + \frac{\sin (B)}{\cos (B)}$

Step 6

Add fractions.

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Step 6.1

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

$\frac{\sin (a)}{\cos (a)} \cdot \frac{\cos (B)}{\cos (B)} + \frac{\sin (B)}{\cos (B)}$

Step 6.2

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

$\frac{\sin (a)}{\cos (a)} \cdot \frac{\cos (B)}{\cos (B)} + \frac{\sin (B)}{\cos (B)} \cdot \frac{\cos (a)}{\cos (a)}$

Step 6.3

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

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Step 6.3.1

Multiply $\frac{\sin (a)}{\cos (a)}$ by $\frac{\cos (B)}{\cos (B)}$ .

$\frac{\sin (a) \cos (B)}{\cos (a) \cos (B)} + \frac{\sin (B)}{\cos (B)} \cdot \frac{\cos (a)}{\cos (a)}$

Step 6.3.2

Multiply $\frac{\sin (B)}{\cos (B)}$ by $\frac{\cos (a)}{\cos (a)}$ .

$\frac{\sin (a) \cos (B)}{\cos (a) \cos (B)} + \frac{\sin (B) \cos (a)}{\cos (B) \cos (a)}$

Step 6.3.3

Reorder the factors of $\cos (a) \cos (B)$ .

$\frac{\sin (a) \cos (B)}{\cos (B) \cos (a)} + \frac{\sin (B) \cos (a)}{\cos (B) \cos (a)}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{\sin (a) \cos (B) + \sin (B) \cos (a)}{\cos (B) \cos (a)}$

Step 7

Reorder terms.

$\frac{\cos (B) \sin (a) + \cos (a) \sin (B)}{\cos (B) \cos (a)}$

Step 8

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

$\frac{\sin (a + B)}{\cos (a) \cos (B)} = \tan (a) + \tan (B)$ is an identity