Question

Prove that (sec$)^2+(cosec$)^2=(sec$)^2×(cosec$)^2

Answer 1

Interpret sec and cosec as sec x = 1/cos x and cosec x = 1/sin x. The identity is

(sec x)^2 + (cosec x)^2 = (sec x)^2 (cosec x)^2.

Write it as
1/cos^2 x + 1/sin^2 x = 1/(cos^2 x sin^2 x).

Multiply both sides by cos^2 x sin^2 x (allowed when sin x and cos x are nonzero) to get

sin^2 x + cos^2 x = 1,

which is the Pythagorean identity and is true. Hence the given identity holds for all x with sin x ≠ 0 and cos x ≠ 0.

(Equivalently, using sec^2 = 1 + tan^2 and cosec^2 = 1 + cot^2: (sec^2−1)(cosec^2−1)=tan^2·cot^2=1, which rearranges to the same result.)