Question

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

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

Answer 1

Answer: (D)

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

$\text{Let}y=\cos x\cos (x+2)-co{s}^2(x+1)\Rightarrow \frac{1}{2}\left[\cos \right(x+x+2\left)\cos \right((x+2)-x\left)\right]-{\cos }^2(x+1)[\text{Using}\cos A.\cos B=\frac{1}{2}[\cos (A+B)+\cos (A-B)\left]\right]$

$\Rightarrow \frac{1}{2}\left[\cos 2\right(x+1)+\cos 2]-{\cos }^{2}2(x+1)$

$\Rightarrow {\cos }^2(x+1)-{\sin }^{2}1-{\cos }^2(x+1)\left[\text{Using}\frac{1}{2}\right[\cos 2A+\cos 2B]=\frac{1}{2}[2{\cos }^{2}A-1+1-2{\sin }^{2}A]={\cos }^{2}A-{\sin }^{2}B]$

$\Rightarrow -{\sin }^{2}1$

This is a straight line which parallel to x axis, it passes through $(\frac{\pi }{2},{\sin }^{2}1)$