Trigonometry Examples

$\sec^{6} (x) (\sec (x) \tan (x)) - \sec^{4} (x) (\sec (x) \tan (x)) = \sec^{5} (x) \tan^{3} (x)$

Step 1

Start on the left side.

$\sec^{6} (x) (\sec (x) \tan (x)) - \sec^{4} (x) (\sec (x) \tan (x))$

Step 2

Simplify each term.

$\sec^{7} (x) \tan (x) - \sec^{5} (x) \tan (x)$

Step 3

Simplify the expression.

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Step 3.1

Factor $\sec^{5} (x) \tan (x)$ out of $\sec^{7} (x) \tan (x) - \sec^{5} (x) \tan (x)$ .

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Step 3.1.1

Factor $\sec^{5} (x) \tan (x)$ out of $\sec^{7} (x) \tan (x)$ .

$\sec^{5} (x) \tan (x) (\sec^{2} (x)) - \sec^{5} (x) \tan (x)$

Step 3.1.2

Factor $\sec^{5} (x) \tan (x)$ out of $- \sec^{5} (x) \tan (x)$ .

$\sec^{5} (x) \tan (x) (\sec^{2} (x)) + \sec^{5} (x) \tan (x) (- 1)$

Step 3.1.3

Factor $\sec^{5} (x) \tan (x)$ out of $\sec^{5} (x) \tan (x) (\sec^{2} (x)) + \sec^{5} (x) \tan (x) (- 1)$ .

$\sec^{5} (x) \tan (x) (\sec^{2} (x) - 1)$

Step 3.2

Apply pythagorean identity.

$\sec^{5} (x) \tan (x) \tan^{2} (x)$

Step 3.3

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

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Step 3.3.1

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

$\sec^{5} (x) (\tan^{2} (x) \tan (x))$

Step 3.3.2

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

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Step 3.3.2.1

Raise $\tan (x)$ to the power of $1$ .

$\sec^{5} (x) (\tan^{2} (x) \tan^{1} (x))$

Step 3.3.2.2

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

$\sec^{5} (x) {\tan (x)}^{2 + 1}$

Step 3.3.3

Add $2$ and $1$ .

$\sec^{5} (x) \tan^{3} (x)$

Step 3.4

Rewrite $\sec (x)$ in terms of sines and cosines.

${(\frac{1}{\cos (x)})}^{5} \tan^{3} (x)$

Step 3.5

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

$\frac{1^{5}}{\cos^{5} (x)} \tan^{3} (x)$

Step 3.6

One to any power is one.

$\frac{1}{\cos^{5} (x)} \tan^{3} (x)$

Step 3.7

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{1}{\cos^{5} (x)} {(\frac{\sin (x)}{\cos (x)})}^{3}$

Step 3.8

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

$\frac{1}{\cos^{5} (x)} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)}$

Step 3.9

Combine.

$\frac{1 \sin^{3} (x)}{\cos^{5} (x) \cos^{3} (x)}$

Step 3.10

Multiply $\cos^{5} (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 3.10.1

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

$\frac{1 \sin^{3} (x)}{{\cos (x)}^{5 + 3}}$

Step 3.10.2

Add $5$ and $3$ .

$\frac{1 \sin^{3} (x)}{\cos^{8} (x)}$

Step 3.11

Multiply $\sin^{3} (x)$ by $1$ .

$\frac{\sin^{3} (x)}{\cos^{8} (x)}$

Step 4

Rewrite $\frac{\sin^{3} (x)}{\cos^{8} (x)}$ as $\sec^{5} (x) \tan^{3} (x)$ .

$\sec^{5} (x) \tan^{3} (x)$

Step 5

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

$\sec^{6} (x) (\sec (x) \tan (x)) - \sec^{4} (x) (\sec (x) \tan (x)) = \sec^{5} (x) \tan^{3} (x)$ is an identity