Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Rewrite $\sec^{4} (x)$ as ${(\sec^{2} (x))}^{2}$ .

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

Step 2.2

Rewrite $\tan^{4} (x)$ as ${(\tan^{2} (x))}^{2}$ .

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sec^{2} (x)$ and $b = \tan^{2} (x)$ .

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

Step 2.4

Simplify each term.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

One to any power is one.

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

Step 2.4.4

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

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

Step 2.4.5

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

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

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

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

Step 3

Now consider the right side of the equation.

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

Step 4

Convert to sines and cosines.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

One to any power is one.

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

Step 6

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

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