Precalculus Examples

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

Step 1

Start on the left side.

$\frac{\sec (x)}{1 + \sec (x)}$

Step 2

Multiply $\frac{\sec (x)}{1 + \sec (x)}$ by $\frac{1 - \sec (x)}{1 - \sec (x)}$ .

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\sec (x)$ by $1$ .

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

Step 4.3

Multiply $\sec (x) (- \sec (x))$ .

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 + \sec (x)) (1 - \sec (x))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply Pythagorean identity.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Apply pythagorean identity.

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

Step 7

Convert to sines and cosines.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

One to any power is one.

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

Step 8.3

To write $\frac{1}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 8.4

Write each expression with a common denominator of ${\cos (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

Add $1$ and $1$ .

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

Step 8.5

Combine the numerators over the common denominator.

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

Step 8.6

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

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Step 8.6.1

Move the leading negative in $- \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ into the numerator.

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

Step 8.6.2

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

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

Step 8.6.3

Cancel the common factor.

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

Step 8.6.4

Rewrite the expression.

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

Step 8.7

Move the negative in front of the fraction.

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

Step 8.8

Apply the distributive property.

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

Step 8.9

Combine $\cos (x)$ and $\frac{1}{{\sin (x)}^{2}}$ .

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

Step 8.10

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Reorder terms.

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

Step 11

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

$\frac{\sec (x)}{1 + \sec (x)} = \frac{1 - \cos (x)}{\sin^{2} (x)}$ is an identity