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.



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