Question

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

Answer 1

$f\left(x\right)=\sin x+[\frac{1}{2}\sin x+\cos x\times \frac{\sqrt{3}}{2}{]}^2+\cos [\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x]$

$\Rightarrow f\left(x\right)=\sin x+\frac{1}{4}[\sqrt{3}\cos x+\sin x{]}^2+\frac{1}{2}[{\cos }^{2}x-\sqrt{3}\sin x\cos x]$

$\Rightarrow {\sin }^{2}x+\frac{3}{4}{\cos }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{\sqrt{3}}{2}\sin x\cos x+\frac{{\cos }^{2}x}{2}-\frac{\sqrt{3}}{2}\sin x\cos x$

$\Rightarrow \frac{5}{4}{\sin }^{2}x+\frac{5}{4}{\cos }^{2}x$

$\Rightarrow \frac{5}{4}({\sin }^{2}x+{\cos }^{2}x)$

$\Rightarrow \frac{5}{4}\therefore f\left(x\right)=\frac{5}{4}$

$\therefore \left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)=g\left(\frac{5}{4}\right)=1$