If $f(x) = \sin^2 x + \sin^2 \left(x +\frac{\pi}{3}\right)+\cos\left(x +\frac{\pi}{2}\right)\cos x$ and $g (5/4) =1$, then $gof (x)$, is

a constant function

The correct answer is Option (2) → a constant function

We have,

$f(x) = \sin^2 x + \sin^2 \left(x +\frac{\pi}{3}\right)+\cos\left(x +\frac{\pi}{2}\right)\cos x$

$⇒f(x) =\frac{1}{2}\left\{1-\cos 2x+1-\cos\left(2x +\frac{2\pi}{3}\right)+\cos\left(2x +\frac{\pi}{3}\right)+\cos x\frac{\pi}{3}\right\}$

$⇒f(x) =\frac{1}{2}\left[\frac{5}{2}-\left\{\cos 2x+\cos\left(2x +\frac{2\pi}{3}\right)\right\}+\cos\left(2x +\frac{\pi}{3}\right)\right]$

$⇒f(x) =\frac{1}{2}\left[\frac{5}{2}-2\cos\left(2x +\frac{\pi}{3}\right)\cos\frac{\pi}{3}+\cos\left(2x +\frac{\pi}{3}\right)\right]$

$⇒f(x) =\frac{5}{4}$ for all $x∈R$

$∴gof(x)=g(f(x))=g(\frac{5}{4})=1$ for all x.

Hence, gof (x) is a constant function.