The equation \(\mathrm{y}=\sin \mathrm{x} \sin (\mathrm{x}+2)-\sin ^{2}(\mathrm{x}+1)\) represents a straight line lying in :

(C) third and fourth quadrants only

(A) first, third and fourth quadrants

(B) first, second and fourth quadrants

(D) second and third quadrants only

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Trigonometric Ratios and Identities

The equation

y = sin x sin (x + 2) - sin^{2} (x + 1)

represents a straight line lying in :

Answer: (C)

Double everything and use product-to-sum identities:

2 y = 2 sin x sin (x + 2) - 2 sin^{2} (x + 1)

. The first term is

cos 2 - cos (2 x + 2)

and the second is

- (1 - cos (2 x + 2))

, so the

cos (2 x + 2)

terms cancel and

2 y = cos 2 - 1

. Thus

y = \frac{cos 2 - 1}{2}

, a constant, and since

cos 2 \approx - 0.42 < 1

this constant is negative. The graph is a horizontal line below the

x

-axis, which runs through the third and fourth quadrants only.

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option