Straight Line and Pair of Straight Lines 1 Question 2
2. The equation $y=\sin x \sin (x+2)-\sin ^{2}(x+1)$ represents a straight line lying in
(a) second and third quadrants only
(b) first, second and fourth quadrants
(c) first, third and fourth quadrants
(d) third and fourth quadrants only
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Solution:
Key Idea Use formulae :
$2 \sin A \sin B=\cos (A-B)-\cos (A+B)$ and $\cos 2 \theta=1-2 \sin ^{2} \theta$
Given equation is $y=\sin x \sin (x+2)-\sin ^{2}(x+1)$
$$ =\frac{1}{2}[\cos 2-\cos (2 x+2)]-\frac{1}{2}[1-\cos (2 x+2)] $$
$[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)$ and $\left.\cos 2 \theta=1-2 \sin ^{2} \theta \Rightarrow 2 \sin ^{2} \theta=1-\cos 2 \theta\right]$
$=\frac{1}{2} \cos 2-\frac{1}{2} \cos (2 x+2)-\frac{1}{2}+\frac{1}{2} \cos (2 x+2)$
$=\frac{1}{2}(\cos (2)-1)=-\frac{1}{2}\left(2 \sin ^{2}(1)\right)$
$=-\sin ^{2}(1)<0 \Rightarrow y<0$
and as we know that $y<0$, is in third and fourth quadrants only.
