Find the Exact Value (sin(theta))/(1+sin(theta))-(sin(theta))/(1-sin(theta))

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

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

Subtract $\sin (θ)$ from $0$ .

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

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

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

Combine exponents.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Move the negative in front of the fraction.

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