Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

One to any power is one.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 4.5

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

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

Step 5

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

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

Step 6

Add fractions.

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Step 6.1

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 6.2

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

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

Step 6.3

Combine the numerators over the common denominator.

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

Step 7

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

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

Step 8

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

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