Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Convert from $\frac{\cos (- x)}{\sin (- x)}$ to $\cot (- x)$ .

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

Step 1.2

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

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

Step 1.3

Rewrite $\frac{\cos (- x)}{\sin (x)}$ as a product.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Divide $\cos (- x)$ by $1$ .

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

Step 1.5.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2

Subtract $\cot (x)$ from $- \cot (x)$ .

$- 2 \cot (x)$