Precalculus Examples

$\frac{\cot^{2} (t)}{\csc (t)} = \frac{1 - \sin^{2} (t)}{\sin (t)}$

Step 1

Start on the left side.

$\frac{\cot^{2} (t)}{\csc (t)}$

Step 2

Apply Pythagorean identity in reverse.

$\frac{\csc^{2} (t) - 1}{\csc (t)}$

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to $\csc (t)$ .

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

Step 3.2

Apply the reciprocal identity to $\csc (t)$ .

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

Step 3.3

Apply the product rule to $\frac{1}{\sin (t)}$ .

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

One to any power is one.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Cancel the common factor of $\sin (t)$ .

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Step 4.4.1

Factor $\sin (t)$ out of ${\sin (t)}^{2}$ .

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 4.5

Rewrite $- 1 \sin (t)$ as $- \sin (t)$ .

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

Step 5

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

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

Step 6

Add fractions.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Combine the numerators over the common denominator.

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

Step 7

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

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

Step 8

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

$\frac{\cot^{2} (t)}{\csc (t)} = \frac{1 - \sin^{2} (t)}{\sin (t)}$ is an identity