UPSC NDA I 2025 Mathematics Paper

Concept:Use trigonometric identities to simplify the sum of tangent and cotangent differences into a single cosecant function.Explanation:Given $p = \tan(2\alpha) - \tan(\alpha)$ and $q = \cot(\alpha) - \cot(2\alpha)$.First, write each term as a single fraction using $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$.Combine $p+q$:$p+q = \frac{\sin(2\alpha)}{\cos(2\alpha)} - \frac{\sin(\alpha)}{\cos(\alpha)} + \frac{\cos(\alpha)}{\sin(\alpha)} - \frac{\cos(2\alpha)}{\sin(2\alpha)}$.Group the first and last terms, and the middle two terms, then simplify using sine and cosine double-angle formulas:$= \frac{\sin^2(2\alpha) - \cos^2(2\alpha)}{\sin(2\alpha)\cos(2\alpha)} + \frac{\cos^2(\alpha) - \sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}$.Using $\cos^2\theta - \sin^2\theta = \cos 2\theta$ and $2\sin\theta\cos\theta = \sin 2\theta$, each fraction becomes $-\frac{2\cos(4\alpha)}{\sin(4\alpha)}$ and $\frac{2\cos(2\alpha)}{\sin(2\alpha)}$.Thus, $p+q = \frac{2\cos(2\alpha)}{\sin(2\alpha)} - \frac{2\cos(4\alpha)}{\sin(4\alpha)}$.Combine over a common denominator: $= \frac{2\big(\sin(4\alpha)\cos(2\alpha) - \cos(4\alpha)\sin(2\alpha)\big)}{\sin(4\alpha)\sin(2\alpha)}$.Apply the sine difference identity: $\sin(4\alpha)\cos(2\alpha) - \cos(4\alpha)\sin(2\alpha) = \sin(4\alpha - 2\alpha) = \sin(2\alpha)$.So $p+q = \frac{2\sin(2\alpha)}{\sin(4\alpha)\sin(2\alpha)} = \frac{2}{\sin(4\alpha)} = 2\csc(4\alpha)$.Answer:Option D: $2\cosec 4\alpha$