The equation
Question:

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

  1. second and third quadrants only

  2. third and fourth quadrants only

  3. first, third and fourth quadrants

  4. first, second and fourth quadrants


Correct Option: , 2

Solution:

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

$2 y=\cos 2-\cos (2 x+2)-(1-\cos (2 x+2))$

$=\cos 2-1$

$2 y=-2 \sin ^{2} \frac{1}{2}$

$y=-\sin ^{2} \frac{1}{2} \leq 0$

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