Question

The graph of the function $\cos x.\cos (x+2)-{\cos }^2(x+1)$ is a

• A. straight line passing through the point $(0,-{\sin }^{2}1)$ with slope $2$
• B. straight line passing through the origin
• C. parabola with vertex $(1,-{\sin }^{2}1)$
• D. straight line passing through the point $(\frac{\pi }{2},-{\sin }^{2}1)$ and parallel to the $x-$axis

Answer 1

Answer: (D)

straight line passing through the point $(\frac{\pi }{2},-{\sin }^{2}1)$ and parallel to the $x-$axis

Let $y=\cos x\cos (x+2)-{\cos }^2(x+1)=\cos (x+1-1)\cos (x+1+1)-{\cos }^2(x+1)$
Using $\frac{1}{2}[\cos (A+B+A-B)+\cos (A+B-A+B)]=\frac{1}{2}[\cos 2A+\cos 2B]=\frac{1}{2}[{\cos }^{2}A-1+1-2{\sin }^{2}A]={\cos }^2(x+1)-{\sin }^{2}1$
$={\cos }^2(x+1)-{\sin }^{2}1-{\cos }^2(x+1)=-{\sin }^{2}1$
This is a straight line which is parallel to x-axis, it passes through $(\frac{\pi }{2},-{\sin }^{2}1)$