Question

lf $f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$ and $g\left(\frac{5}{4}\right)=1$, $g\left(1\right)=0$ then $\left(gof\right)\left(x\right)=$

• A. $1$
• B. $0$
• C. $\sin x$
• D. Data is insufficient

Answer 1

The correct option is A $1$

$f\left(x\right)={\sin }^{2}x+{(\sin x\cos \frac{\pi }{3}+\cos x\sin \frac{\pi }{3})}^2+cosx\left(\cos x\cos \frac{\pi }{3}-\sin x\sin \frac{\pi }{3}\right)$
$={\sin }^{2}x+{[\frac{\sin x}{2}+\frac{\sqrt{3}\cos x}{2}]}^2+\frac{{\cos }^{2}x}{2}-\frac{\sqrt{3}}{2}\cos x\sin x$
$={\sin }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{3}{4}{\cos }^{2}x+\frac{\sqrt{3}}{2}\sin x\cos x+\frac{{\cos }^{2}x}{2}-\frac{\sqrt{3}}{2}\sin x\cos x$
$=\frac{5}{4}({\sin }^{2}x+{\cos }^{2}x)=\frac{5}{4}$
$\therefore$ $\left[gof\right]\left(x\right)=g\left[f\left(x\right)\right]=g\left(\frac{5}{4}\right)=1$