Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

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

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

Step 4

Subtract fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply Pythagorean identity.

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Step 6.1

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

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

Step 6.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 6.3

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 6.4

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 6.5

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 6.6

Apply pythagorean identity.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Write $- 1$ as a fraction with denominator $1$ .

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

Step 9

Combine.

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

Step 10

Multiply $\cos (x)$ by $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

Now consider the right side of the equation.

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

Step 13

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

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

Step 14

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

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

Step 15

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

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