Algebra Examples

$\tan^{2} (x) + 4 = \sec^{2} (x) + 3$

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

Add $- 1$ and $4$ .

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

Step 5

Reorder terms.

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

Step 6

Rewrite $3 + \frac{1}{\cos^{2} (x)}$ as $\sec^{2} (x) + 3$ .

$\sec^{2} (x) + 3$

Step 7

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

$\tan^{2} (x) + 4 = \sec^{2} (x) + 3$ is an identity