Equivalent Representations of Trigonometric Functions

The same trig expression can be written in many different ways, just as $x^{2} - 9$ and $(x + 3)(x - 3)$ are the same polynomial. An identity is an equation that is true for every input where both sides are defined. This note (Part A) builds the Pythagorean identities and uses them to simplify expressions, prove identities, and solve equations. Part B adds the sum and double-angle identities.