Question

Prove that: (sin$)^2(Cot$^2(cosec$)^2(tan$)^2=1

Answer 1

Let θ be the angle. Using the reciprocal relationships csc θ = 1/sin θ and cot θ = 1/tan θ (or cot θ = cos θ / sin θ, tan θ = sin θ / cos θ), we have

sin θ · csc θ = 1, and tan θ · cot θ = 1.

Squaring both equalities gives sin^2 θ · csc^2 θ = 1 and tan^2 θ · cot^2 θ = 1. Multiplying these two equalities yields

sin^2 θ · cot^2 θ · csc^2 θ · tan^2 θ = 1.

(Valid for angles where all functions are defined, i.e. sin θ ≠ 0 and cos θ ≠ 0 — equivalently θ ≠ kπ/2 for any integer k.)