Precalculus Examples

$\frac{\cos (w)}{\cos (w) + \sin (w)} = \frac{\cot (w)}{1 + \cot (w)}$

Step 1

Start on the right side.

$\frac{\cot (w)}{1 + \cot (w)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{\frac{\cos (w)}{\sin (w)}}{1 + \cot (w)}$

Step 2.2

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

$\frac{\frac{\cos (w)}{\sin (w)}}{1 + \frac{\cos (w)}{\sin (w)}}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (w)}{\sin (w)} \cdot \frac{1}{1 + \frac{\cos (w)}{\sin (w)}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$\frac{\cos (w)}{\sin (w)} \cdot \frac{1}{\frac{\sin (w)}{\sin (w)} + \frac{\cos (w)}{\sin (w)}}$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{\cos (w)}{\sin (w)} \cdot \frac{1}{\frac{\sin (w) + \cos (w)}{\sin (w)}}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (w)}{\sin (w)} (1 \frac{\sin (w)}{\sin (w) + \cos (w)})$

Step 3.4

Multiply $\frac{\sin (w)}{\sin (w) + \cos (w)}$ by $1$ .

$\frac{\cos (w)}{\sin (w)} \cdot \frac{\sin (w)}{\sin (w) + \cos (w)}$

Step 3.5

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

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Step 3.5.1

Cancel the common factor.

$\frac{\cos (w)}{\sin (w)} \cdot \frac{\sin (w)}{\sin (w) + \cos (w)}$

Step 3.5.2

Rewrite the expression.

$\cos (w) \frac{1}{\sin (w) + \cos (w)}$

Step 3.6

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

$\frac{\cos (w)}{\sin (w) + \cos (w)}$

Step 4

Reorder terms.

$\frac{\cos (w)}{\cos (w) + \sin (w)}$

Step 5

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

$\frac{\cos (w)}{\cos (w) + \sin (w)} = \frac{\cot (w)}{1 + \cot (w)}$ is an identity