If $f(x)=\sin^2 x + \sin^2(x +\frac{π}{3}) + \cos(x +\frac{π}{3})\cos x$ and $g (5/4) =1$, then $gof (x)$, is
Answer & explanation
Correct answer: option 2
We have,
$f(x)=\sin^2 x + \sin^2(x +\frac{π}{3}) + \cos(x +\frac{π}{3})\cos x$
$⇒f(x)=\frac{1}{2}\left.\begin{matrix}1-\cos 2x+1-\cos(2x+\frac{2π}{3})+\cos(2x+\frac{π}{3})+\frac{π}{3}\cos\end{matrix}\right\}$
$⇒f(x)=\frac{1}{2}\left[\frac{5}{2}-\left.\begin{matrix}\cos 2x+\cos(2x+\frac{2π}{3})\end{matrix}\right\}+\cos(2x+\frac{π}{3})\right]$
$⇒f(x)=\frac{1}{2}\left[\frac{5}{2}-2\cos(2x+\frac{π}{3})\cos\frac{π}{3}+\cos(2x+\frac{π}{3})\right]$
$⇒f(x)=\frac{5}{4}$ for all $x ∈ R$.
$∴gof(x)=g(f(x))=g(\frac{5}{4})=1$ for all x.
Hence, $gof (x)$ is a constant function.