Trigonometry Examples

$\frac{\tan^{3} (x) (x) - \sec^{2} (x) x \tan (x)}{\cot (- x)}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

$\frac{\tan (x) x (\tan^{2} (x)) - \sec^{2} (x) x \tan (x)}{\cot (- x)}$

Step 1.1.2

Factor $\tan (x) x$ out of $- \sec^{2} (x) x \tan (x)$ .

$\frac{\tan (x) x (\tan^{2} (x)) + \tan (x) x (- \sec^{2} (x))}{\cot (- x)}$

Step 1.1.3

Factor $\tan (x) x$ out of $\tan (x) x (\tan^{2} (x)) + \tan (x) x (- \sec^{2} (x))$ .

$\frac{\tan (x) x (\tan^{2} (x) - \sec^{2} (x))}{\cot (- x)}$

Step 1.2

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

$\frac{\tan (x) x (- \sec^{2} (x) + \tan^{2} (x))}{\cot (- x)}$

Step 1.3

Factor $- 1$ out of $- \sec^{2} (x)$ .

$\frac{\tan (x) x (- (\sec^{2} (x)) + \tan^{2} (x))}{\cot (- x)}$

Step 1.4

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

$\frac{\tan (x) x (- (\sec^{2} (x)) - 1 (- \tan^{2} (x)))}{\cot (- x)}$

Step 1.5

Factor $- 1$ out of $- (\sec^{2} (x)) - 1 (- \tan^{2} (x))$ .

$\frac{\tan (x) x (- (\sec^{2} (x) - \tan^{2} (x)))}{\cot (- x)}$

Step 1.6

Apply pythagorean identity.

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

Step 1.7

Combine exponents.

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Step 1.7.1

Factor out negative.

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

Step 1.7.2

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

$\frac{- (x \cdot \tan (x))}{\cot (- x)}$

$\frac{- x \cdot \tan (x)}{\cot (- x)}$

Step 2

Since $\cot (- x)$ is an odd function, rewrite $\cot (- x)$ as $- \cot (x)$ .

$\frac{- x \cdot \tan (x)}{- \cot (x)}$

Step 3

Dividing two negative values results in a positive value.

$\frac{x \cdot \tan (x)}{\cot (x)}$

Step 4

Separate fractions.

$\frac{x}{1} \cdot \frac{\tan (x)}{\cot (x)}$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

$\frac{x}{1} (\tan (x) \tan (x))$

Step 8.3

Raise $\tan (x)$ to the power of $1$ .

$\frac{x}{1} (\tan^{1} (x) \tan (x))$

Step 8.4

Raise $\tan (x)$ to the power of $1$ .

$\frac{x}{1} (\tan^{1} (x) \tan^{1} (x))$

Step 8.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x}{1} {\tan (x)}^{1 + 1}$

Step 8.6

Add $1$ and $1$ .

$\frac{x}{1} \tan^{2} (x)$

Step 9

Divide $x$ by $1$ .

$x \tan^{2} (x)$