How to simplify a rational trigonometric expression by factoring

The video tutorial explains how to simplify a rational expression involving trigonometric functions. The instructor discusses different methods, including using the conjugate and Pythagorean identity, but opts for a more straightforward approach by factoring the expression. By recognizing the trinomial form, the expression is factored and simplified, resulting in a concise solution.

What reasoning does the teacher provide for not wanting to multiply the binomial by the trinomial?

What is the final simplified expression that the teacher arrives at?