Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

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

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Step 2.1

Multiply by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Apply Pythagorean identity.

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Step 3.1

Rearrange terms.

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

Step 3.2

Apply pythagorean identity.

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Simplify.

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Step 6.1

One to any power is one.

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

Step 6.2

One to any power is one.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Combine.

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

Step 6.5

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

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

Step 6.6

Multiply $1$ by $1$ .

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

Add $2$ and $2$ .

$\frac{1}{{\cos (x)}^{4}} - \frac{1}{{\cos (x)}^{2}}$

Step 6.8

To write $- \frac{1}{{\cos (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Step 6.9

Write each expression with a common denominator of ${\cos (x)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.9.1

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

$\frac{1}{{\cos (x)}^{4}} - \frac{{\cos (x)}^{2}}{{\cos (x)}^{2} {\cos (x)}^{2}}$

Step 6.9.2

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

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Step 6.9.2.1

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

$\frac{1}{{\cos (x)}^{4}} - \frac{{\cos (x)}^{2}}{{\cos (x)}^{2 + 2}}$

Step 6.9.2.2

Add $2$ and $2$ .

$\frac{1}{{\cos (x)}^{4}} - \frac{{\cos (x)}^{2}}{{\cos (x)}^{4}}$

Step 6.10

Combine the numerators over the common denominator.

$\frac{1 - {\cos (x)}^{2}}{{\cos (x)}^{4}}$

Step 6.11

Simplify the numerator.

$\frac{(1 + \cos (x)) (1 - \cos (x))}{\cos^{4} (x)}$

Step 7

Rewrite $\frac{(1 + \cos (x)) (1 - \cos (x))}{\cos^{4} (x)}$ as $\sec^{4} (x) - \sec^{2} (x)$ .

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

Step 8

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

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