Assuming you mean verify, not simplify, let's proceed.
1/(1-costheta)+1/(1+costheta)=2csc^2theta
When verifying an equation, it is good practice to only work one side of the equation and turn it into the other side. Let's turn the left side into the right side.
First, multiply each fraction by a clever form of 1, noting that (a+b)(a-b)=a^2-b^2.
1/(1-costheta)color(green)(times(1+costheta)/(1+costheta))+1/(1+costheta)color(green)(times(1-costheta)/(1-costheta))=(1+costheta)/(1-cos^2theta)+(1-costheta)/(1-cos^2theta)
Now combine like terms.
(1+costheta+1-costheta)/(1-cos^2theta)=2/(1-cos^2theta)
The Pythagorean identity states that sin^2theta+cos^2theta=1. This can be rearranged to sin^2theta=1-cos^2theta, so our expression is equal to 2/sin^2theta.
Since csctheta=1/sintheta, 1/sin^2theta=csc^2theta so our expression is equal to 2csc^2theta, which is what we wanted to prove.
