Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Add fractions.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 6

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

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

Step 7

Apply Pythagorean identity.

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Step 7.1

Reorder $- \cos^{2} (x)$ and $1$ .

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

Step 7.2

Apply pythagorean identity.

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

Step 8

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

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

Step 9

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

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