Trigonometry Examples

$\tan^{2} (x) \csc^{2} (x) = \sec^{2} (x)$

Step 1

Start on the left side.

$\tan^{2} (x) \csc^{2} (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

${(\frac{\sin (x)}{\cos (x)})}^{2} \csc^{2} (x)$

Step 2.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 2.3

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

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

Step 2.4

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 3

Simplify.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of ${\sin (x)}^{2}$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

One to any power is one.

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

Step 4

Rewrite $\frac{1}{\cos^{2} (x)}$ as $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 5

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

$\tan^{2} (x) \csc^{2} (x) = \sec^{2} (x)$ is an identity