Algebra Examples

$\tan^{3} (x) \csc^{2} (x) - \tan^{3} (x)$

Step 1

Reorder $\tan^{3} (x) \csc^{2} (x)$ and $- \tan^{3} (x)$ .

$- \tan^{3} (x) + \tan^{3} (x) \csc^{2} (x)$

Step 2

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

$\tan^{3} (x) \cdot - 1 + \tan^{3} (x) \csc^{2} (x)$

Step 3

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

$\tan^{3} (x) \cdot - 1 + \tan^{3} (x) (\csc^{2} (x))$

Step 4

Factor $\tan^{3} (x)$ out of $\tan^{3} (x) \cdot - 1 + \tan^{3} (x) (\csc^{2} (x))$ .

$\tan^{3} (x) \cdot (- 1 + \csc^{2} (x))$

Step 5

Rearrange terms.

$\tan^{3} (x) \cdot (\csc^{2} (x) - 1)$

Step 6

Apply pythagorean identity.

$\tan^{3} (x) \cdot \cot^{2} (x)$

Step 7

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

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

Step 8

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

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

Step 9

Apply basic rules of exponents.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Cancel the common factor of $\sin^{2} (x)$ .

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Step 10.1

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

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Cancel the common factor of $\cos^{2} (x)$ .

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Step 11.1

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

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

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

$\tan (x)$