Precalculus Examples

$\frac{\csc (4 x) - \cot (4 x)}{\csc (4 x) + \cot (4 x)}$

Step 1

Replace the expressions with an equivalent expression using the fundamental identities.

$\frac{1 (\frac{1}{\sin (4 x)} - \frac{\cos (4 x)}{\sin (4 x)})}{\frac{1}{\sin (4 x)} + \frac{\cos (4 x)}{\sin (4 x)}}$

Step 2

Simplify.

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Step 2.1

Combine into one fraction.

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Step 2.1.1

Multiply $\frac{1}{\sin (4 x)} - \frac{\cos (4 x)}{\sin (4 x)}$ by $1$ .

$\frac{\frac{1}{\sin (4 x)} - \frac{\cos (4 x)}{\sin (4 x)}}{\frac{1}{\sin (4 x)} + \frac{\cos (4 x)}{\sin (4 x)}}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{\frac{1}{\sin (4 x)} - \frac{\cos (4 x)}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Convert from $\frac{\cos (4 x)}{\sin (4 x)}$ to $\cot (4 x)$ .

$\frac{\frac{1}{\sin (4 x)} - \cot (4 x)}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.2

To write $- \cot (4 x)$ as a fraction with a common denominator, multiply by $\frac{\sin (4 x)}{\sin (4 x)}$ .

$\frac{\frac{1}{\sin (4 x)} - \cot (4 x) \cdot \frac{\sin (4 x)}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.3

Combine $- \cot (4 x)$ and $\frac{\sin (4 x)}{\sin (4 x)}$ .

$\frac{\frac{1}{\sin (4 x)} + \frac{- \cot (4 x) \sin (4 x)}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{\frac{1 - \cot (4 x) \sin (4 x)}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.5

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.2.5.1

Add parentheses.

$\frac{\frac{1 - (\cot (4 x) \sin (4 x))}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.5.2

Rewrite $- \cot (4 x) \sin (4 x)$ in terms of sines and cosines.

$\frac{\frac{1 - (\frac{\cos (4 x)}{\sin (4 x)} \sin (4 x))}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.2.5.3

Cancel the common factors.

$\frac{\frac{1 - \cos (4 x)}{\sin (4 x)}}{\frac{1 + \cos (4 x)}{\sin (4 x)}}$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 - \cos (4 x)}{\sin (4 x)} \cdot \frac{\sin (4 x)}{1 + \cos (4 x)}$

Step 2.4

Cancel the common factor of $\sin (4 x)$ .

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Step 2.4.1

Cancel the common factor.

$\frac{1 - \cos (4 x)}{\sin (4 x)} \cdot \frac{\sin (4 x)}{1 + \cos (4 x)}$

Step 2.4.2

Rewrite the expression.

$(1 - \cos (4 x)) \frac{1}{1 + \cos (4 x)}$

Step 2.5

Apply the distributive property.

$1 \frac{1}{1 + \cos (4 x)} - \cos (4 x) \frac{1}{1 + \cos (4 x)}$

Step 2.6

Multiply $\frac{1}{1 + \cos (4 x)}$ by $1$ .

$\frac{1}{1 + \cos (4 x)} - \cos (4 x) \frac{1}{1 + \cos (4 x)}$

Step 2.7

Combine $\frac{1}{1 + \cos (4 x)}$ and $\cos (4 x)$ .

$\frac{1}{1 + \cos (4 x)} - \frac{\cos (4 x)}{1 + \cos (4 x)}$

Step 2.8

Combine the numerators over the common denominator.

$\frac{1 - \cos (4 x)}{1 + \cos (4 x)}$