Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify each term.

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

Step 5

Apply the distributive property.

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

Step 6

Simplify each term.

$\frac{1 - \cos^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}$

Step 7

Apply pythagorean identity.

$\frac{\sin^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}$

Step 8

Reorder terms.

$\frac{\cos^{4} (x) + \sin^{2} (x)}{\cos^{2} (x)}$

Step 9

Now consider the right side of the equation.

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

Step 10

Convert to sines and cosines.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Write $\cos^{2} (x)$ as a fraction with denominator $1$ .

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

Step 12

Add fractions.

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Step 12.1

To write $\frac{\cos^{2} (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

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

Step 12.2

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

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

Step 12.3

Combine the numerators over the common denominator.

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

Step 13

Multiply ${\cos (x)}^{2}$ by ${\cos (x)}^{2}$ by adding the exponents.

$\frac{\cos^{4} (x) + \sin^{2} (x)}{\cos^{2} (x)}$

Step 14

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

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