Trigonometry Examples

$\frac{\tan^{2} (x)}{\sec (x)} = \sec (x) - \cos (x)$

Start on the left side.

$\frac{\tan^{2} (x)}{\sec (x)}$

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

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

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

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

One to any power is one.

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

Apply the distributive property.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Now consider the right side of the equation.

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

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

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

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

$\frac{\tan^{2} (x)}{\sec (x)} = \sec (x) - \cos (x)$ is an identity