Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Add fractions.

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Step 3.1

To write $\frac{- \sin (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.2

Multiply $\frac{- \sin (x)}{1}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 4

Multiply $- \sin (x) \sin (x)$ .

$\frac{1 - \sin^{2} (x)}{\sin (x)}$

Step 5

Apply pythagorean identity.

$\frac{\cos^{2} (x)}{\sin (x)}$

Step 6

Rewrite $\frac{\cos^{2} (x)}{\sin (x)}$ as $\cot (x) \cos (x)$ .

$\cot (x) \cos (x)$

Step 7

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

$\frac{1}{\sin (x)} - \sin (x) = \cot (x) \cos (x)$ is an identity