The primary idea behind establishing a trigonometry identity is to transform the Left Hand Side (LHS) to the Right Hand Side (RHS) or the other way around. One way to verify the trig identity that you see below is to multiply the LHS by a fraction that has the conjugage in the numerator and the denominator of the fraction.
What is the conjugate? The conjugate is what you get if you reverse the sign in the middle of a binomial. For example the conjugate of is
So if you multiply the LHS by a fraction that has the conjugate as the numerator and the denominator, you will get numerator that can be rewritten using one of the Pythagorean Identities.
The following shows you step by step how to verify the trig Identity.