CDS 2023 PYQ with Answer — If tan x = (sin θ + cos θ)/(sin θ − cos θ), π/4 < θ < π/2, then what…

2023 · Q28

If tan x = (sin θ + cos θ)/(sin θ − cos θ), π/4 < θ < π/2, then what is √2 sin x equal to?

Elementary MathematicsTrigonometry

Asin θ + cos θ
Bsin θ − cos θ
C(sin θ + cos θ)/√2
D(sin θ − cos θ)/√2

Explanation

Divide numerator and denominator by cosθ: tan x = (1 + tanθ)/(1 − tanθ) = tan(π/4 + θ), using tan(A+B) = (tanA + tanB)/(1 − tanA·tanB) with A = π/4. Hence x = θ + π/4 (within the admissible range). Then √2·sin x = √2·sin(θ + π/4) = √2[sinθ·cos(π/4) + cosθ·sin(π/4)] = √2·(sinθ + cosθ)/√2 = sinθ + cosθ. The answer is (A).