Trigonometry Examples

$\frac{\csc (- x)}{\sec (- x)} = - \cot (x)$

Step 1

Start on the left side.

$\frac{\csc (- x)}{\sec (- x)}$

Step 2

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

$\frac{- \csc (x)}{\sec (- x)}$

Step 3

Since $\sec (- x)$ is an even function, rewrite $\sec (- x)$ as $\sec (x)$ .

$\frac{- \csc (x)}{\sec (x)}$

Step 4

Convert to sines and cosines.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 6

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

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

Step 7

Combine.

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

Step 8

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

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Now consider the right side of the equation.

$- \cot (x)$

Step 11

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

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

Step 12

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

$\frac{\csc (- x)}{\sec (- x)} = - \cot (x)$ is an identity