Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 4

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

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