Trigonometry Examples

$\tan^{2} (θ) = \sin^{2} (θ) \cdot \sec^{2} (θ)$

Step 1

Start on the right side.

$\sin^{2} (θ) \cdot \sec^{2} (θ)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

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

Step 3.2

Combine ${\sin (θ)}^{2}$ and $\frac{1}{{\cos (θ)}^{2}}$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$

Step 4

Rewrite $\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ as $\tan^{2} (θ)$ .

$\tan^{2} (θ)$

Step 5

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

$\tan^{2} (θ) = \sin^{2} (θ) \cdot \sec^{2} (θ)$ is an identity