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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Add $1$ and $1$ .

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

Subtract $\cos (θ)$ from $\cos (θ)$ .

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

Add $2$ and $0$ .

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