Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Rewrite $\csc (2 x)$ in terms of sines and cosines.

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

Step 2

Simplify the right side.

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Step 2.1

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

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Step 2.1.1

Separate fractions.

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

Step 2.1.2

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

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

Step 2.1.3

Rewrite $\frac{\frac{1}{\sin (x)}}{\cos (x)}$ as a product.

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

Step 2.1.4

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 2.1.5

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

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

Step 3

Multiply both sides of the equation by $\sin (2 x)$ .

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

Step 4

Cancel the common factor of $\sin (2 x)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Apply the sine double-angle identity.

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$1 = \frac{\cos (x)}{\cos (x)}$

Step 9

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

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Step 9.1

Cancel the common factor.

$1 = \frac{\cos (x)}{\cos (x)}$

Step 9.2

Rewrite the expression.

$1 = 1$

Step 10

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 11

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$