Question

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

• A. 1
• B. 0
• C. $\sin x$
• D. $noneofthese$

Answer 1

Answer: (A)

$f\left(x\right)={\sin }^{2}x+{\sin }^2\left(x+\frac{\pi }{3}\right)$
$+\cos x\cos \left(x+\frac{\pi }{3}\right)$ and $g\left(\frac{5}{4}\right)=1$
$f\left(x\right)={\sin }^{2}x+{\left(\sin x\cos \frac{\pi }{3}+\cos x\sin \frac{\pi }{3}\right)}^2$
$+\cos x\left(\cos x\cos \frac{x}{3}-\sin x\sin \frac{\pi }{3}\right)$
$={\sin }^{2}x{\left(\frac{\sin x}{2}+\frac{\sqrt{3}}{2}\cos x\right)}^2$
$+\cos x\left(\frac{\cos x}{2}-\frac{\sqrt{3}}{2}\sin x\right)$
$={\sin }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{3}{4}{\cos }^{2}x$
$+\frac{\sqrt{3}}{2}\sin x\cos x\frac{{\cos }^{2}x}{2}\Rightarrow =\frac{\sqrt{3}}{2}sinxcosx$
$=\frac{5}{4}si{n}^{2}x+\frac{5}{4}co{s}^{2}xf\left(x\right)=\frac{5}{4}$
$\therefore \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(f\left(x\right)\right)=g\left(\frac{5}{4}\right)=1$.