Precalculus Examples

$\frac{\cot^{3} (x)}{\csc (x)}$

Step 1

Separate fractions.

$\frac{\cot^{3} (x)}{1} \cdot \frac{1}{\csc (x)}$

Step 2

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

$\frac{\cot^{3} (x)}{1} \cdot \frac{1}{\frac{1}{\sin (x)}}$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{\cot^{3} (x)}{1} (1 \sin (x))$

Step 4

Multiply $\sin (x)$ by $1$ .

$\frac{\cot^{3} (x)}{1} \sin (x)$

Step 5

Divide $\cot^{3} (x)$ by $1$ .

$\cot^{3} (x) \sin (x)$

Step 6

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

${(\frac{\cos (x)}{\sin (x)})}^{3} \sin (x)$

Step 7

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\cos^{3} (x)}{\sin^{3} (x)} \sin (x)$

Step 8

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

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Step 8.1

Factor $\sin (x)$ out of $\sin^{3} (x)$ .

$\frac{\cos^{3} (x)}{\sin (x) \sin^{2} (x)} \sin (x)$

Step 8.2

Cancel the common factor.

$\frac{\cos^{3} (x)}{\sin (x) \sin^{2} (x)} \sin (x)$

Step 8.3

Rewrite the expression.

$\frac{\cos^{3} (x)}{\sin^{2} (x)}$

Step 9

Factor $\cos^{2} (x)$ out of $\cos^{3} (x)$ .

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

Step 10

Multiply by $1$ .

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

Step 11

Separate fractions.

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

Step 12

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$\frac{\cos (x)}{1} \cot^{2} (x)$

Step 13

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

$\cos (x) \cot^{2} (x)$