Precalculus Examples

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

Step 1

Start on the left side.

$(\sec^{2} (θ) - 1) \cos^{2} (θ)$

Step 2

Apply pythagorean identity.

$\tan^{2} (θ) \cos^{2} (θ)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

$\sin^{2} (θ)$

Step 5

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

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