Precalculus Examples

\sec^{2} (x) (1 - \sin^{2} (x)) = 1

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

Step 1

Start on the left side.

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

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

Step 2

Apply pythagorean identity.

\sec^{2} (x) \cos^{2} (x)

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

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to

\sec (x)

$\sec (x)$ .

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

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

Step 3.2

Apply the product rule to

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

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

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

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

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

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

Step 4.2

Cancel the common factor of

{\cos (x)}^{2}

${\cos (x)}^{2}$ .

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Step 4.2.1

Cancel the common factor.

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

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

Step 4.2.2

Rewrite the expression.

1

$1$

1

$1$

1

$1$

Step 5

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

\sec^{2} (x) (1 - \sin^{2} (x)) = 1

$\sec^{2} (x) (1 - \sin^{2} (x)) = 1$ is an identity

\sec^{2} x (1 - \sin^{2} x) = 1

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

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