UPSC CDS 2 2021 Math Paper

Concept:Factor out common trigonometric functions and use $\sin^2\theta + \cos^2\theta = 1$ to simplify the expression.Explanation:Start with the given expression:$$\frac{2\sin^3\theta - \sin\theta}{\cos\theta - 2\cos^3\theta}$$Factor $\sin\theta$ from the numerator and $\cos\theta$ from the denominator:$$= \frac{\sin\theta (2\sin^2\theta - 1)}{\cos\theta (1 - 2\cos^2\theta)}$$Now replace $1$ with $\sin^2\theta + \cos^2\theta$ in both numerator and denominator:$$= \frac{\sin\theta (2\sin^2\theta - (\sin^2\theta + \cos^2\theta))}{\cos\theta ((\sin^2\theta + \cos^2\theta) - 2\cos^2\theta)}$$Simplify inside the brackets:Numerator: $2\sin^2\theta - \sin^2\theta - \cos^2\theta = \sin^2\theta - \cos^2\theta$Denominator: $\sin^2\theta + \cos^2\theta - 2\cos^2\theta = \sin^2\theta - \cos^2\theta$Thus the expression becomes:$$= \frac{\sin\theta (\sin^2\theta - \cos^2\theta)}{\cos\theta (\sin^2\theta - \cos^2\theta)}$$Cancel the common factor $(\sin^2\theta - \cos^2\theta)$ (non‑zero for $0° < \theta<90°$ except $\theta=45°$; the simplified form holds as a limit):$$= \frac{\sin\theta}{\cos\theta} = \tan\theta$$Answer:$\boxed{\tan\theta}$, which corresponds to option C.