Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Move $- 1$ .

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

Step 1.4

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 1.5

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.6

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 1.7

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 1.8

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

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

Step 1.9

Apply pythagorean identity.

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

Step 1.10

Factor $\sin^{2} (x)$ out of $- \sin^{2} (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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Step 1.10.1

Factor $\sin^{2} (x)$ out of $- \sin^{2} (x)$ .

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

Step 1.10.2

Factor $\sin^{2} (x)$ out of $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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

Step 1.10.3

Factor $\sin^{2} (x)$ out of $\sin^{2} (x) \cdot - 1 + \sin^{2} (x) \frac{1}{\cos^{2} (x)}$ .

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{\cos (x)}$ and $b = 1$ .

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

Step 2

Cancel the common factor of $\sin^{2} (x)$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(\frac{1}{\cos (x)} + 1) (\frac{1}{\cos (x)} - 1)$ by $1$ .

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

Step 3

Expand $(\frac{1}{\cos (x)} + 1) (\frac{1}{\cos (x)} - 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

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

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

Step 4.1.1.5

Add $1$ and $1$ .

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

Step 4.1.2

Combine $\frac{1}{\cos (x)}$ and $- 1$ .

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

Step 4.1.3

Move the negative in front of the fraction.

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

Step 4.1.4

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

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

Step 4.1.5

Multiply $- 1$ by $1$ .

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

Step 4.2

Add $- \frac{1}{\cos (x)}$ and $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos^{2} (x)} + 0 - 1$

Step 4.3

Add $\frac{1}{\cos^{2} (x)}$ and $0$ .

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Apply pythagorean identity.

$\tan^{2} (x)$