The equation y = s i n x sin ⁡ x + 2 - sin 2 ⁡ ( x + 1 ) represents a straight line lying in:

The equation y=sinxsin⁡x+2-sin2⁡(x+1) represents a straight line lying in:
  1. first, third and fourth quadrants
  2. second and third quadrants only
  3. first, second and fourth quadrants
  4. third and fourth quadrants only

Solution

Here's the corrected text with proper LaTeX formatting: $y = \sin(x+2)\sin(x) - \sin^2(x+1)$ $= \frac{1}{2} \left[ 2\sin(x+2)\sin(x) - 2\sin^2(x+1) \right]$ $\left[ 2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B) \quad \& \quad \cos^2(A) = 1 - 2\sin^2(A) \right]$ $= \frac{1}{2} \left[ \cos(x+2-x) - \cos(2x+2) - 1 + 2\cos^2(x+1) \right]$ $= \frac{1}{2} \left[ \cos(2) - \cos(2x+2) - 1 + \cos(2(x+1)) \right]$ $= \frac{1}{2} \left[ 1 - \cos(2) \right] = -\frac{1}{2}2\sin^2(1)$ $= -\sin^2(1) \quad (\text{constant and negative})$ Hence, the straight line lies in the third and fourth quadrants only.

Asked in: JEE Main 2019 (12 Apr Shift 1)

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