Precalculus Examples

$(1 + \tan^{2} (θ)) \cos^{2} (θ) = 1$

Step 1

Start on the left side.

$(1 + \tan^{2} (θ)) \cos^{2} (θ)$

Step 2

Apply Pythagorean identity.

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Step 2.1

Rearrange terms.

$(\tan^{2} (θ) + 1) \cos^{2} (θ)$

Step 2.2

Apply pythagorean identity.

$\sec^{2} (θ) \cos^{2} (θ)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$1$

Step 5

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

$(1 + \tan^{2} (θ)) \cos^{2} (θ) = 1$ is an identity