Trigonometry Examples

$2 \cot (x) = \csc^{2} (x) \sin (2 x)$

Step 1

Simplify $\csc^{2} (x) \sin (2 x)$ .

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Step 1.1

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

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 1.2.2

One to any power is one.

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

Step 1.3

Combine $\frac{1}{\sin^{2} (x)}$ and $\sin (2 x)$ .

$2 \cot (x) = \frac{\sin (2 x)}{\sin^{2} (x)}$

Step 1.4

Apply the sine double-angle identity.

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

Step 1.5

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

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Step 1.5.1

Factor $\sin (x)$ out of $2 \sin (x) \cos (x)$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

Separate fractions.

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

Step 1.7

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

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

Step 1.8

Divide $2$ by $1$ .

$2 \cot (x) = 2 \cot (x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Step 3