Question

The number of integer(s) in the range of $f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$ is

Answer 1

$f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$
$={\sin }^{2}x+1-{\cos }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$
$=1+{\sin }^{2}x+\cos (x+\frac{\pi }{3})[\cos x-\cos (x+\frac{\pi }{3})]$
$=1+{\sin }^{2}x+\cos (x+\frac{\pi }{3})\left[2\sin \frac{\pi }{6}\sin (x+\frac{\pi }{6})\right]$
$=1+{\sin }^{2}x+\frac{1}{2}\left[2\cos (x+\frac{\pi }{3})\sin (x+\frac{\pi }{6})\right]$
$=1+{\sin }^{2}x+\frac{1}{2}[\sin (2x+\frac{\pi }{2})-\sin \frac{\pi }{6}]=1+{\sin }^{2}x+\frac{1}{2}[\cos 2x-\frac{1}{2}]=1+{\sin }^{2}x+\frac{1}{2}[1-2{\sin }^{2}x-\frac{1}{2}]$
$\Rightarrow f\left(x\right)=\frac{5}{4}$
Since, $f\left(x\right)$ is constant function,
hence, its range contains only one element i.e., $\frac{5}{4}.$