Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) + \tan^{4} (x)$ .

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Step 2.1

Multiply by $1$ .

$(\tan^{2} (x) \cdot 1 + \tan^{4} (x)) \sec^{2} (x)$

Step 2.2

Factor $\tan^{2} (x)$ out of $\tan^{4} (x)$ .

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

Step 2.3

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

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

Step 3

Apply Pythagorean identity.

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Step 3.1

Rearrange terms.

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

Step 3.2

Apply pythagorean identity.

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

Step 4

Convert to sines and cosines.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

Simplify.

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Step 5.1

One to any power is one.

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

Step 5.2

Combine.

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

Step 5.3

Combine.

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

Step 5.4

Multiply $1^{2}$ by $1$ by adding the exponents.

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Step 5.4.1

Move $1$ .

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

Step 5.4.2

Multiply $1$ by $1^{2}$ .

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Step 5.4.2.1

Raise $1$ to the power of $1$ .

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

Step 5.4.2.2

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

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

Step 5.4.3

Add $1$ and $2$ .

$\frac{{\sin (x)}^{2} \cdot 1^{3}}{{\cos (x)}^{2} {\cos (x)}^{2} {\cos (x)}^{2}}$

Step 5.5

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

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Step 5.5.1

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

$\frac{{\sin (x)}^{2} \cdot 1^{3}}{{\cos (x)}^{2 + 2} {\cos (x)}^{2}}$

Step 5.5.2

Add $2$ and $2$ .

$\frac{{\sin (x)}^{2} \cdot 1^{3}}{{\cos (x)}^{4} {\cos (x)}^{2}}$

Step 5.6

Multiply ${\cos (x)}^{4}$ by ${\cos (x)}^{2}$ by adding the exponents.

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Step 5.6.1

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

$\frac{{\sin (x)}^{2} \cdot 1^{3}}{{\cos (x)}^{4 + 2}}$

Step 5.6.2

Add $4$ and $2$ .

$\frac{{\sin (x)}^{2} \cdot 1^{3}}{{\cos (x)}^{6}}$

Step 5.7

One to any power is one.

$\frac{{\sin (x)}^{2} \cdot 1}{{\cos (x)}^{6}}$

Step 5.8

Multiply ${\sin (x)}^{2}$ by $1$ .

$\frac{\sin^{2} (x)}{\cos^{6} (x)}$

Step 6

Rewrite $\frac{\sin^{2} (x)}{\cos^{6} (x)}$ as $\sec^{4} (x) \tan^{2} (x)$ .

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

Step 7

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

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