Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply pythagorean identity.

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Convert to sines and cosines.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Simplify.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2

One to any power is one.

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

Step 5.3

One to any power is one.

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

Step 5.4

Subtract $1$ from $- 1$ .

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

Step 5.5

Divide ${\cos (x)}^{2}$ by $1$ .

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

Step 5.6

Apply the distributive property.

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

Step 5.7

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

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

Step 6

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

$1 - 2 \cos^{2} (x) = \frac{\tan^{2} (x) - 1}{\tan^{2} (x) + 1}$ is an identity