Q.Prove that: $(1-\cos^2\theta)(1+\cot^2\theta) = 1$.

Both brackets simplify using one of the three standard identities each, and the two simplified factors are reciprocals of each other.

Step 1 — simplify the first bracket using the Pythagorean identity.

Step 2 — simplify the second bracket using the derived identity.

Step 3 — multiply the simplified brackets. …

An equally valid route rewrites everything in sines and cosines from the start: $1-\cos^2\theta=\sin^2\theta$ and $1+\cot^2\theta = 1+\dfrac{\cos^2\theta}{\sin^2\theta}=\dfrac{\sin^2\theta+\c …

A common error is trying to prove this by plugging in a single standard angle (say $\theta=45°$) and stopping there — checking one value only confirms the identity for that angle, not for ever …