Trigonometry Examples

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

Step 1

Start on the left side.

$\tan^{2} (θ) \cos^{2} (θ)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

$\sin^{2} (θ)$

Step 4

Rewrite $\sin^{2} (θ)$ as $\frac{1}{\csc^{2} (θ)}$ .

$\frac{1}{\csc^{2} (θ)}$

Step 5

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

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