Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

One to any power is one.

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

Step 4.3

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

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

Step 4.4

Apply the distributive property.

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

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

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