UPSC CDS II 2024 Math Paper

Concept:Simplify the trigonometric expression by using identities and substitution.Explanation:Step 1: Write the second factor in terms of $\sin\theta$ and $\cos\theta$:$\cot\theta - \csc\theta = \frac{\cos\theta}{\sin\theta} - \frac{1}{\sin\theta} = \frac{\cos\theta - 1}{\sin\theta}$.Step 2: Divide the numerator and denominator of the first factor by $\sin\theta$ (assuming $\sin\theta \neq 0$):$\frac{\cos\theta - \sin\theta + 1}{\cos\theta + \sin\theta - 1} = \frac{\cot\theta - 1 + \csc\theta}{\cot\theta + 1 - \csc\theta}$.Step 3: Let $x = \cot\theta - \csc\theta$. Then note the identity:$(\cot\theta + \csc\theta)(\cot\theta - \csc\theta) = \cot^2\theta - \csc^2\theta = -1$.Thus $\cot\theta + \csc\theta = -\frac{1}{x}$.Step 4: Substitute into the first factor:Numerator: $\cot\theta - 1 + \csc\theta = (\cot\theta + \csc\theta) - 1 = -\frac{1}{x} - 1 = -\frac{1+x}{x}$.Denominator: $\cot\theta + 1 - \csc\theta = (\cot\theta - \csc\theta) + 1 = x + 1$.Therefore the first factor becomes $\frac{-\frac{1+x}{x}}{x+1} = -\frac{1}{x}$ (since $x+1 \neq 0$ cancels).Step 5: Multiply the two factors:$E = \left(-\frac{1}{x}\right) \cdot x = -1$.Thus the entire expression simplifies to $-1$, independent of $\theta$ (except where undefined).Answer:$-1$ (Option A).