Trigonometry

• sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.
• sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.
• Compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B).
• Double angle formulae for sin 2A, cos 2A (three forms) and tan 2A.
• a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ ± α), with R = √(a² + b²).

The other three functions

sec, cosec and cot are reciprocals, so they are undefined wherever cos, sin or tan is zero; their graphs have vertical asymptotes there. sec θ and cosec θ are never between −1 and 1. Divide sin²θ + cos²θ ≡ 1 by cos²θ or sin²θ to get the two new identities.

Expanding, doubling and combining

The compound angle formulae give exact values for angles like 15° and 75°, and simplify expressions such as cos(x − 30°). Put A = B to get the double angle formulae. Choose the form of cos 2A that leaves one trig function in the equation.

To write a cos θ + b sin θ as R cos(θ − α), expand R cos(θ − α) = R cos θ cos α + R sin θ sin α and compare: R cos α = a, R sin α = b. Then R = √(a² + b²) and tan α = b/a. This form gives the greatest and least values (±R) and turns the equation into a single cosine.