Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

$\frac{\csc (- x)}{\cot (- x)} \cdot \frac{\tan (- x)}{\frac{1}{\cos (- x)}}$

Step 3

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

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (- x)}$ .

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

Step 5

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

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

Step 6

Cancel the common factor of $\cos (- x)$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{\csc (- x)}{\cot (- x)} \sin (- x)$

Step 7

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

$\frac{- \csc (x)}{\cot (- x)} \sin (- x)$

Step 8

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

$\frac{- \csc (x)}{- \cot (x)} \sin (- x)$

Step 9

Dividing two negative values results in a positive value.

$\frac{\csc (x)}{\cot (x)} \sin (- x)$

Step 10

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

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

Step 11

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

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

Step 12

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 13

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

Tap for more steps...

Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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

Step 14

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

$\sec (x) \sin (- x)$

Step 15

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

$\sec (x) (- \sin (x))$

Step 16

Rewrite using the commutative property of multiplication.

$- \sec (x) \sin (x)$