UPSC CDS 1 2022 Math Paper

Concept:Use trigonometric simplification or substitution to relate the two expressions.Explanation:Method 1 (Quick substitution):Take $\theta = 45°$.Then $x = \frac{1 - \cos 45° + \sin 45°}{1 + \sin 45°} = \frac{1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} = \frac{\sqrt{2}}{\sqrt{2}+1}$.Simplify: $x = 2 - \sqrt{2}$.Now compute $y = \frac{\sin \theta + \cos \theta - 1}{\cos \theta}$ with the same $\theta = 45°$:$y = \frac{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1}{\frac{1}{\sqrt{2}}} = 2 - \sqrt{2}$.Thus $x = y$, so the required expression equals $x$.Method 2 (Algebraic manipulation):Start with $x = \frac{1 - \cos\theta + \sin\theta}{1 + \sin\theta}$.Rationalize by multiplying numerator and denominator by $(1 - \sin\theta)$:$x = \frac{(1 - \cos\theta + \sin\theta)(1 - \sin\theta)}{(1 + \sin\theta)(1 - \sin\theta)} = \frac{1 - \sin\theta - \cos\theta + \cos\theta\sin\theta + \sin\theta - \sin^2\theta}{1 - \sin^2\theta}$.Simplify numerator: $1 - \cos\theta + \cos\theta\sin\theta - \sin^2\theta$.Since $1 - \sin^2\theta = \cos^2\theta$, we have:$x = \frac{\cos^2\theta - \cos\theta + \cos\theta\sin\theta}{\cos^2\theta}$.Factor $\cos\theta$ from numerator and denominator:$x = \frac{\cos\theta - 1 + \sin\theta}{\cos\theta}$.This is exactly $\frac{\sin\theta + \cos\theta - 1}{\cos\theta}$.Therefore the required value is $x$.Answer:$x$ (Option B)