Trigonometry Examples

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

Step 1

Multiply both sides by $\csc^{2} (x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

Simplify the expression.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

One to any power is one.

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Apply the sine double-angle identity.

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

Step 2.1.1.5

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

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Step 2.1.1.5.1

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

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

Step 2.1.1.5.2

Cancel the common factors.

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Step 2.1.1.5.2.1

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

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

Step 2.1.1.5.2.2

Cancel the common factor.

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

Step 2.1.1.5.2.3

Rewrite the expression.

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

Step 2.1.1.6

Separate fractions.

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

Step 2.1.1.7

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

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

Step 2.1.1.8

Divide $2$ by $1$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $\cot (x)$ to the left side of the equation.

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Step 3.1.1

Subtract $2 \cot (x)$ from both sides of the equation.

$2 \cot (x) - 2 \cot (x) = 0$

Step 3.1.2

Subtract $2 \cot (x)$ from $2 \cot (x)$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$