How do I simplify (1 / [1 + cos(x) ] ) + (1 / [1 - cos(x) ] )?

In order to add the fractions together, we must have a common denominator of the fractions. The simplest way to do this is to make the denominators of the equations the product of the original two denomniators. In other words, the denominator for both fractions should be  (1 + cos(x) ) (1 - cos(x) ). To do this, we can multiply each fraction by an equation equal to 1 which gives the correct denominator (because an equation multiplied by 1 is equal to the original equation). Hence, we multiply the first fraction by (1 - cos(x) ) / (1 - cos(x) ), and the second by (1 + cos(x) ) / (1 + cos(x) ). This will allow us to add both fractions together as we would with non-algebraic fractions and gives us (1 - cos(x) + 1 + cos(x) ) / ( [1 + cos x] [1 - cos x] )  which is simplified to 2 / (1 - cos²(x) ). We should remember the identity 1 - cos²(x) = sin²(x) from earlier in the course but the proof is trivial if it is needed. So, our equation can be simplified to 2 / sin²(x). From here you should be able to see this is equal to 2 * (1 / sin²(x)) which is the same as 2 * cosec²(x). This cannot be simplified any further so our final answer is 2cosec²(x).