UPSC NDA I 2018 Mathematics Solved Paper

Concept:Use the componendo and dividendo rule after expanding the sine ratios.Explanation:Given: $\frac{\sin (x + y)}{\sin (x - y)} = \frac{a + b}{a - b}$Expand using addition formulas:$\frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} = \frac{a + b}{a - b}$Apply componendo and dividendo (add and subtract numerator and denominator on both sides):$\frac{(\sin x \cos y + \cos x \sin y) + (\sin x \cos y - \cos x \sin y)}{(\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y)} = \frac{(a + b) + (a - b)}{(a + b) - (a - b)}$Simplify each part:Left side: numerator = $2 \sin x \cos y$, denominator = $2 \cos x \sin y$Right side: numerator = $2a$, denominator = $2b$Thus: $\frac{2 \sin x \cos y}{2 \cos x \sin y} = \frac{2a}{2b}$Cancel the common factor 2: $\frac{\sin x \cos y}{\cos x \sin y} = \frac{a}{b}$Rewrite as $\frac{\tan x}{\tan y} = \frac{a}{b}$Answer:$\frac{\tan x}{\tan y} = \frac{a}{b}$, which corresponds to option A.