Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Since

\csc (- x)

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

\csc (- x)

$\csc (- x)$ as

- \csc (x)

$- \csc (x)$ .

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

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

Step 3

Since

\sec (- x)

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

\sec (- x)

$\sec (- x)$ as

\sec (x)

$\sec (x)$ .

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

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

Step 4

Convert to sines and cosines.

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Step 4.1

Apply the reciprocal identity to

\csc (x)

$\csc (x)$ .

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

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

Step 4.2

Apply the reciprocal identity to

\sec (x)

$\sec (x)$ .

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

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

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

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

Step 5

Simplify.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 5.2

Combine

\cos (x)

$\cos (x)$ and

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

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

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

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

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

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

Step 6

Write

- 1

$- 1$ as a fraction with denominator

1

$1$ .

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

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

Step 7

Combine.

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

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

Step 8

Multiply

\sin (x)

$\sin (x)$ by

1

$1$ .

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

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

Step 9

Move the negative in front of the fraction.

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

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

Step 10

Now consider the right side of the equation.

- \cot (x)

$- \cot (x)$

Step 11

Write

\cot (x)

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

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

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

Step 12

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

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

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