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. $1$
• C. $\sin 1^o$
• D. None of these

Answer 1

Answer: (B)

$1$

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