Trigonometry Examples

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

Step 1

Apply pythagorean identity.

$\frac{2 \cot (x)}{\csc^{2} (x)}$

Step 2

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

$\frac{2 \cot (x)}{\csc (x) \csc (x)}$

Step 3

Separate fractions.

$\frac{2}{\csc (x)} \cdot \frac{\cot (x)}{\csc (x)}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$\frac{2}{\csc (x)} \cos (x)$

Step 9

Separate fractions.

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the expression.

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Step 12.1

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

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

Step 12.2

Divide $2$ by $1$ .

$2 \sin (x) \cos (x)$

Step 13

Apply the sine double-angle identity.

$\sin (2 x)$