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

Both bracketed factors are themselves standard forms of a Pythagorean identity, so rewriting each in terms of $\sin\theta$ collapses the product to $1$.

Step 1 — rewrite the first factor. From $\sin^2\theta+\cos^2\theta=1$, $1-\cos^2\theta=\sin^2\theta$.

Step 2 — rewrite the second factor. From the identity $1+\cot^2\theta=\operatorname{cosec}^2\theta$, the second bracket is exactly $\operatorname{cosec}^2\theta$.

Step 3 — multiply. $\sin^2\theta\cdot\operatorname{cosec}^2\theta = \sin^2\theta\cdot\dfrac{1}{\sin^2\theta}=1$, which is the right-hand side, so the identity is proved. …

Alternatively, expand $1+\cot^2\theta=1+\dfrac{\cos^2\theta}{\sin^2\theta}=\dfrac{\sin^2\theta+\cos^2\theta}{\sin^2\theta}=\dfrac{1}{\sin^2\theta}$ directly from the definition of $\cot\theta$, without quoting the identity by name, then multi …

Trying to expand $(1-\cos^2\theta)(1+\cot^2\theta)$ term-by-term in $\cos\theta$ and $\cot\theta$ instead of first recognising the standard substitutions — technically possi …