Trigonometry Examples

$\csc (- x) - \sin (- x) = - \cos (x) \cot (x)$

Step 1

Start on the left side.

$\csc (- x) - \sin (- x)$

Step 2

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

$- \csc (x) - \sin (- x)$

Step 3

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$- \csc (x) - - \sin (x)$

Step 4

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

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

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

Simplify each term.

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

Step 8

Apply Pythagorean identity.

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Step 8.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 8.2

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 8.3

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 8.4

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 8.5

Apply pythagorean identity.

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Write $- 1$ as a fraction with denominator $1$ .

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

Step 11

Combine.

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

Step 12

Multiply $\sin (x)$ by $1$ .

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

Step 13

Move the negative in front of the fraction.

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

Step 14

Now consider the right side of the equation.

$- \cos (x) \cot (x)$

Step 15

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 16

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

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

Step 17

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

$\csc (- x) - \sin (- x) = - \cos (x) \cot (x)$ is an identity