Precalculus Examples

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

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factor of $\cos (x)$ .

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Step 4.1.3.1

Factor $\cos (x)$ out of $- {\cos (x)}^{2}$ .

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

Step 4.1.3.2

Cancel the common factor.

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

Step 4.1.3.3

Rewrite the expression.

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

Step 4.2

Add $- \cos (x)$ and $\cos (x)$ .

$\frac{1}{\cos (x)} + 0$

Step 4.3

Add $\frac{1}{\cos (x)}$ and $0$ .

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

Step 5

Rewrite $\frac{1}{\cos (x)}$ as $\sec (x)$ .

$\sec (x)$

Step 6

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

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