Algebra Examples

$\frac{\tan (y)}{\csc (y)} = \frac{1}{\cos (y)} - \frac{1}{\sec (y)}$

Step 1

Start on the right side.

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Add fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply pythagorean identity.

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

Step 7

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

$\frac{\tan (y)}{\csc (y)}$

Step 8

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

$\frac{\tan (y)}{\csc (y)} = \frac{1}{\cos (y)} - \frac{1}{\sec (y)}$ is an identity