Trigonometry Examples

$\tan^{4} (x) = \tan^{2} (x) \sec^{2} (x) - \tan^{2} (x)$

Step 1

Start on the right side.

$\tan^{2} (x) \sec^{2} (x) - \tan^{2} (x)$

Step 2

Apply Pythagorean identity.

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Factor $\tan^{2} (x)$ out of $\tan^{2} (x) \sec^{2} (x)$ .

$\tan^{2} (x) (\sec^{2} (x)) - \tan^{2} (x)$

Factor $\tan^{2} (x)$ out of $- \tan^{2} (x)$ .

$\tan^{2} (x) (\sec^{2} (x)) + \tan^{2} (x) \cdot - 1$

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) (\sec^{2} (x)) + \tan^{2} (x) \cdot - 1$ .

$\tan^{2} (x) (\sec^{2} (x) - 1)$

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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

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

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

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

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

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

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

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

Step 4

Simplify.

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Combine.

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

Multiply ${\sin (x)}^{2}$ by ${\sin (x)}^{2}$ by adding the exponents.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $2$ .

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{2} {\cos (x)}^{2}}$

Multiply ${\cos (x)}^{2}$ by ${\cos (x)}^{2}$ by adding the exponents.

$\frac{\sin^{4} (x)}{\cos^{4} (x)}$

Step 5

Rewrite $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ as $\tan^{4} (x)$ .

$\tan^{4} (x)$

Step 6

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

$\tan^{4} (x) = \tan^{2} (x) \sec^{2} (x) - \tan^{2} (x)$ is an identity