Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Simplify the numerator.

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

Step 4

Now consider the right side of the equation.

$\sec (x) + \cos (x)$

Step 5

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

$\frac{1}{\cos (x)} + \cos (x)$

Step 6

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{1}{\cos (x)} + \frac{\cos (x)}{1}$

Step 7

Add fractions.

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Step 7.1

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

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

Step 7.2

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

$\frac{1}{\cos (x)} + \frac{\cos (x) \cos (x)}{\cos (x)}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{1 + \cos (x) \cos (x)}{\cos (x)}$

Step 8

Multiply $\cos (x) \cos (x)$ .

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

Step 9

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

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