Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 1.1.4

Cancel the common factor of $\sin (x)$ .

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Step 1.1.4.1

Cancel the common factor.

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

Step 1.1.4.2

Rewrite the expression.

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

Step 1.1.5

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

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

Step 1.1.6

One to any power is one.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2

Apply pythagorean identity.

$\tan^{2} (x)$