Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2

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

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

Step 3

Dividing two negative values results in a positive value.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x)$