Question

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

• A. first, second and fourth quadrants
• B. second and third quadrants only
• C. third and fourth quadrants only
• D. first, third and fourth quadrants

Answer 1

The correct option is C third and fourth quadrants only

$y=\frac{1}{2}[2\sin x\cdot \sin (x+2)-2{\sin }^2(x+1\left)\right]$
$\Rightarrow y=\frac{1}{2}[\cos 2-\cos (2x+2)-1+\cos (2x+2\left)\right]$
[Since, $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ and ${\sin }^{2}A=1-\cos 2A$]
$\Rightarrow y=\frac{1}{2}(\cos 2-1)=-{\sin }^{2}1<0$
Hence the straight line lying in third and fourth quadrants only.