Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 2.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

One to any power is one.

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

One to any power is one.

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

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

Step 3.2.2

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

$\frac{1}{{\sin (x)}^{2}} \cdot \frac{1}{\frac{1}{{\sin (x)}^{2}} + \frac{- 2 {\sin (x)}^{2}}{{\sin (x)}^{2}}}$

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.3

Combine.

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

Step 3.4

Multiply $1$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Reduce the expression by cancelling the common factors.

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

Step 4

Apply the cosine double-angle identity.

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

Step 5

Rewrite $\frac{1}{\cos (2 x)}$ as $\sec (2 x)$ .

$\sec (2 x)$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sec (2 x) = \frac{\csc^{2} (x)}{\csc^{2} (x) - 2}$ is an identity