Write the Fraction in Simplest Form 1/(1-sin(theta))+1/(1+sin(theta))

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Add $1$ and $1$ .

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

Subtract $\sin (θ)$ from $\sin (θ)$ .

$\frac{2 + 0}{(1 + \sin (θ)) (1 - \sin (θ))}$

Add $2$ and $0$ .

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