Trigonometry Examples

$\sec (B) - \cos (B) = \sin (B) \tan (B)$

Step 1

Start on the left side.

$\sec (B) - \cos (B)$

Step 2

Apply the reciprocal identity to $\sec (B)$ .

$\frac{1}{\cos (B)} - \cos (B)$

Step 3

Write $- \cos (B)$ as a fraction with denominator $1$ .

$\frac{1}{\cos (B)} + \frac{- \cos (B)}{1}$

Step 4

Add fractions.

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Step 4.1

To write $\frac{- \cos (B)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos (B)}{\cos (B)}$ .

$\frac{1}{\cos (B)} + \frac{- \cos (B)}{1} \cdot \frac{\cos (B)}{\cos (B)}$

Step 4.2

Multiply $\frac{- \cos (B)}{1}$ by $\frac{\cos (B)}{\cos (B)}$ .

$\frac{1}{\cos (B)} + \frac{- \cos (B) \cos (B)}{\cos (B)}$

Step 4.3

Combine the numerators over the common denominator.

$\frac{1 - \cos (B) \cos (B)}{\cos (B)}$

Step 5

Multiply $- \cos (B) \cos (B)$ .

$\frac{1 - \cos^{2} (B)}{\cos (B)}$

Step 6

Apply pythagorean identity.

$\frac{\sin^{2} (B)}{\cos (B)}$

Step 7

Rewrite $\frac{\sin^{2} (B)}{\cos (B)}$ as $\sin (B) \tan (B)$ .

$\sin (B) \tan (B)$

Step 8

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

$\sec (B) - \cos (B) = \sin (B) \tan (B)$ is an identity