Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

One to any power is one.

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

One to any power is one.

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

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

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Step 2.5.1

Move the leading negative in $- \frac{1}{\cos^{2} (x)}$ into the numerator.

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

Step 2.5.2

Cancel the common factor.

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

Step 2.5.3

Rewrite the expression.

$2 \cos^{2} (x) - 1$

Step 2.6

Apply the cosine double-angle identity.

$\cos (2 x)$

Step 3

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

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