Trigonometry Examples

$\frac{\sec (x)}{\tan (x)} - \frac{\tan (x)}{\sec (x)} = \cos (x) \cot (x)$

Step 1

Start on the left side.

$\frac{\sec (x)}{\tan (x)} - \frac{\tan (x)}{\sec (x)}$

Step 2

Simplify each term.

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Step 2.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (x)}}{\frac{\sin (x)}{\cos (x)}} - \frac{\tan (x)}{\sec (x)}$

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.4

Cancel the common factor of $\cos (x)$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$\frac{1}{\sin (x)} - \frac{\tan (x)}{\sec (x)}$

Step 2.5

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 2.6

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 2.8

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

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

Step 2.9

Cancel the common factor of $\cos (x)$ .

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Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Add fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply pythagorean identity.

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

Step 7

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

$\cos (x) \cot (x)$

Step 8

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

$\frac{\sec (x)}{\tan (x)} - \frac{\tan (x)}{\sec (x)} = \cos (x) \cot (x)$ is an identity