Trigonometry Examples

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

Step 1

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

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

Step 2

Separate fractions.

$\frac{2}{\cot (x - 1)} \cdot \frac{\cot (x)}{\cot (x - 1)}$

Step 3

Rewrite $\cot (x - 1)$ in terms of sines and cosines.

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Convert from $\frac{\sin (x - 1)}{\cos (x - 1)}$ to $\tan (x - 1)$ .

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

Step 7

Separate fractions.

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

Step 8

Rewrite $\cot (x - 1)$ in terms of sines and cosines.

$\frac{2}{1} \cdot \frac{1}{\frac{\cos (x - 1)}{\sin (x - 1)}} (\cot (x) \tan (x - 1))$

Step 9

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

$\frac{2}{1} \cdot \frac{\sin (x - 1)}{\cos (x - 1)} (\cot (x) \tan (x - 1))$

Step 10

Convert from $\frac{\sin (x - 1)}{\cos (x - 1)}$ to $\tan (x - 1)$ .

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

Step 11

Divide $2$ by $1$ .

$2 \tan (x - 1) (\cot (x) \tan (x - 1))$

Step 12

Multiply $2 \tan (x - 1) (\cot (x) \tan (x - 1))$ .

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Add $1$ and $1$ .

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