Trigonometry Examples

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

Step 1

Start on the right side.

$\sec (x) - \cos (x)$

Step 2

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

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

Step 3

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

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

Step 4

Add fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply pythagorean identity.

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

Step 7

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

$\sin (x) \tan (x)$

Step 8

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

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