Algebra Examples

$\frac{\sec (x) - 1}{\sec (x) + 1} = \frac{A - \cos (x)}{B + \cos (x)}$

Step 1

Rewrite the equation as $\frac{A - \cos (x)}{B + \cos (x)} = \frac{\sec (x) - 1}{\sec (x) + 1}$ .

$\frac{A - \cos (x)}{B + \cos (x)} = \frac{\sec (x) - 1}{\sec (x) + 1}$

Step 2

Multiply both sides by $B + \cos (x)$ .

$\frac{A - \cos (x)}{B + \cos (x)} (B + \cos (x)) = \frac{\sec (x) - 1}{\sec (x) + 1} (B + \cos (x))$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $B + \cos (x)$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{A - \cos (x)}{B + \cos (x)} (B + \cos (x)) = \frac{\sec (x) - 1}{\sec (x) + 1} (B + \cos (x))$

Step 3.1.1.2

Rewrite the expression.

$A - \cos (x) = \frac{\sec (x) - 1}{\sec (x) + 1} (B + \cos (x))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{\sec (x) - 1}{\sec (x) + 1} (B + \cos (x))$ .

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Step 3.2.1.1

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

$A - \cos (x) = \frac{\frac{1}{\cos (x)} - 1}{\sec (x) + 1} (B + \cos (x))$

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Combine.

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

Step 3.2.1.4

Apply the distributive property.

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

Step 3.2.1.5

Simplify by cancelling.

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Step 3.2.1.5.1

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

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Step 3.2.1.5.1.1

Cancel the common factor.

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

Step 3.2.1.5.1.2

Rewrite the expression.

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

Step 3.2.1.5.2

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

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Step 3.2.1.5.2.1

Cancel the common factor.

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

Step 3.2.1.5.2.2

Rewrite the expression.

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

Step 3.2.1.6

Simplify the numerator.

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Step 3.2.1.6.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.2.1.6.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.2.1.7

Simplify terms.

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Step 3.2.1.7.1

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

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

Step 3.2.1.7.2

Apply the distributive property.

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

Step 3.2.1.7.3

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

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

Step 3.2.1.7.4

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

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

Step 3.2.1.7.5

Combine the numerators over the common denominator.

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

Step 3.2.1.7.6

Factor $1 - \cos (x)$ out of $(1 - \cos (x)) B + (1 - \cos (x)) \cos (x)$ .

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Step 3.2.1.7.6.1

Factor $1 - \cos (x)$ out of $(1 - \cos (x)) B$ .

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

Step 3.2.1.7.6.2

Factor $1 - \cos (x)$ out of $(1 - \cos (x)) \cos (x)$ .

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

Step 3.2.1.7.6.3

Factor $1 - \cos (x)$ out of $(1 - \cos (x)) (B) + (1 - \cos (x)) (\cos (x))$ .

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

Step 4

Add $\cos (x)$ to both sides of the equation.

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