Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

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

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

Step 4.3

One to any power is one.

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

Step 4.4

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

$\cos^{2} (x)$

Step 5

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

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