Precalculus Examples

$\frac{1}{1 - \cos (θ)} - \frac{1}{1 + \cos (θ)} = 2 \cot (θ) \csc (θ)$

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $(1 - \cos (θ)) (1 + \cos (θ))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(1 - \cos (θ)) (1 + \cos (θ))$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

Multiply $- - \cos (θ)$ .

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

Multiply $\cos (θ)$ by $1$ .

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

Step 3.2

Subtract $1$ from $1$ .

$\frac{0 + \cos (θ) + \cos (θ)}{(1 + \cos (θ)) (1 - \cos (θ))}$

Step 3.3

Add $0$ and $\cos (θ)$ .

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

Step 3.4

Add $\cos (θ)$ and $\cos (θ)$ .

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

Step 4

Simplify denominator.

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Step 4.1

Expand $(1 + \cos (θ)) (1 - \cos (θ))$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

$\frac{2 \cos (θ)}{1 - \cos^{2} (θ)}$

Step 5

Apply pythagorean identity.

$\frac{2 \cos (θ)}{\sin^{2} (θ)}$

Step 6

Rewrite $\frac{2 \cos (θ)}{\sin^{2} (θ)}$ as $2 \cot (θ) \csc (θ)$ .

$2 \cot (θ) \csc (θ)$

Step 7

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

$\frac{1}{1 - \cos (θ)} - \frac{1}{1 + \cos (θ)} = 2 \cot (θ) \csc (θ)$ is an identity