Question

If $f\left(x\right)=\left[\cos x\cos \right(x+2)-{\cos }^2(x+1\left)\right]$ where $[.]$ denotes the greatest integer function$\le x$. Then solution of the equation $f\left(x\right)=x$ is :

• A. $1$
• B. $-1$
• C. $0$
• D. none of these

Answer 1

Answer: (D)

none of these

As $\cos x\cos (x+2)-{\cos }^2(x+1)$
$=\frac{1}{2}(\cos (2x+2)+\cos 2)-\left(\frac{\cos (2x+2)-1}{2}\right)$
$=\frac{\cos 2}{2}-\frac{1}{2}=-{\sin }^{2}1$
Hence $f\left(x\right)=[-{\sin }^{2}1]=1$ has infinite solutions