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 $(g\circ f)\left(x\right)$ is equal to

• A. $0$
• B. $2$
• C. $1$
• D. $3$

Answer 1

Answer: (C)

$1$

Given, $f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$
$f\left(x\right)={\sin }^{2}x+{(\sin x\cos \frac{\pi }{3}+\cos x\sin \frac{\pi }{3})}^2+\cos x\cos (x+\frac{\pi }{3})$
$\Rightarrow f\left(x\right)={\sin }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{3{\cos }^{2}x}{4}+\frac{2\sqrt{3}}{2.2}\times \sin x\cos x+\frac{{\cos }^{2}x}{2}-\cos x\sin x\frac{\sqrt{3}}{2}$
$=\frac{5}{4}{\sin }^{2}x+\frac{5}{4}{\cos }^{2}x=\frac{5}{4}$
and $g\circ f\left(x\right)=g\left(f\right(x\left)\right)=g\left(\frac{5}{4}\right)=1$