Algebra Examples

$\frac{(\tan^{2} (y) + 1)}{\sec (y)} = \sec (y)$

Step 1

Start on the left side.

$\frac{(\tan^{2} (y) + 1)}{\sec (y)}$

Step 2

Apply pythagorean identity.

$\frac{\sec^{2} (y)}{\sec (y)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

$\frac{{(\frac{1}{\cos (y)})}^{2}}{\frac{1}{\cos (y)}}$

Step 3.3

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

$\frac{\frac{1^{2}}{\cos^{2} (y)}}{\frac{1}{\cos (y)}}$

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1^{2}}{{\cos (y)}^{2}} \cos (y)$

Step 4.2

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

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Step 4.2.1

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

$\frac{1^{2}}{\cos (y) \cos (y)} \cos (y)$

Step 4.2.2

Cancel the common factor.

$\frac{1^{2}}{\cos (y) \cos (y)} \cos (y)$

Step 4.2.3

Rewrite the expression.

$\frac{1^{2}}{\cos (y)}$

Step 4.3

One to any power is one.

$\frac{1}{\cos (y)}$

Step 5

Rewrite $\frac{1}{\cos (y)}$ as $\sec (y)$ .

$\sec (y)$

Step 6

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

$\frac{\tan^{2} (y) + 1}{\sec (y)} = \sec (y)$ is an identity