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 = sin/cos, cot = cos/sin) and reciprocal identities (csc = 1/sin, sec = 1/cos, cot = 1/tan). Mastery of these allows rapid simplification of seemingly complex expressions.