Question

$f\left(x\right)={sin}^{2}x+si{n}^2(x+\frac{\pi }{3})+cos(x+\frac{\pi }{3})cosx$ and $g\left(\frac{5}{4}\right)=1$ then $g\left(f\right(x\left)\right)$ is

• A. A polynomial of first degree in $\sin x$ and $\cos x$
• B. A constant function
• C. A polynomial of second degree in $\sin x$ and $\cos x$
• D. None of these

Answer 1

The correct option is B A constant function

Given,

${\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos (x+\frac{\pi }{3})\cos x$

$={\sin }^{2}x+{[\frac{1}{2}\sin x+\cos x\frac{\sqrt{3}}{2}]}^2+\cos x[\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x]$

$={\sin }^{2}x+\frac{1}{4}{[\sqrt{3}\cos x+\sin x]}^2+\frac{1}{2}[{\cos }^{2}x-\sqrt{3}\sin x\cos x]$

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

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

$=\frac{5}{4}[{\sin }^{2}x+{\cos }^{2}x]$

$=\frac{5}{4}$

we have $g\left(\frac{5}{4}\right)=1$

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

Hence it is a constant function.