The three Pythagorean identities form the backbone of trigonometric simplification: $sin^2$(x) + $cos^2$(x) = 1, 1 + $tan^2$(x) = $sec^2$(x), and 1 + $cot^2$(x) = $csc^2$(x). The second and third are obtained by dividing the first by $cos^2$(x) and $sin^2$(x) respectively. These identities are used to: (1) eliminate one ratio in favor of another, (2) simplify complex expressions, (3) prove other identities. Additional fundamental identities include the quotient identities (tan = $\frac{sin}{cos}$, cot = $\frac{cos}{sin}$) and reciprocal identities (csc = 1/sin, sec = 1/cos, cot = 1/tan). Mastery of these allows rapid simplification of seemingly complex expressions.