Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $\sin (θ)$ out of $\sin (θ) (1 - \sin (θ)) - \sin (θ) (1 + \sin (θ))$ .

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Step 5.1.1

Factor $\sin (θ)$ out of $- \sin (θ) (1 + \sin (θ))$ .

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

Step 5.1.2

Factor $\sin (θ)$ out of $\sin (θ) (1 - \sin (θ)) + \sin (θ) (- (1 + \sin (θ)))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Subtract $1$ from $1$ .

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

Step 5.5

Subtract $\sin (θ)$ from $0$ .

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

Step 5.6

Subtract $\sin (θ)$ from $- \sin (θ)$ .

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

Step 5.7

Combine exponents.

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Step 5.7.1

Raise $\sin (θ)$ to the power of $1$ .

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

Step 5.7.2

Raise $\sin (θ)$ to the power of $1$ .

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

Step 5.7.3

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

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

Step 5.7.4

Add $1$ and $1$ .

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

Step 6

Move the negative in front of the fraction.

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