# If f(x)=sin^2 x+sin^2 (x+pi/3) +cos x cos(x+pi/3) and g(5/4)=1, then gof(pi/8) =?

Solution:   We have ,

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

$\Rightarrow$      $f(x)=\frac{1}{2}(1-\cos 2x)+\frac{1}{2}[1-\cos(2x+\frac{2\pi}{3})]+\frac{1}{2}[\cos(2x+\frac{\pi}{3})+\cos \frac{\pi}{3}]$

$\Rightarrow$     $f(x)=\frac{5}{4}-\frac{1}{2}[\cos 2x+\cos (2x+\frac{2\pi}{3})-\cos(2x+\frac{\pi}{3})]$

$f(x)=\frac{5}{4}-\frac{1}{2}[2\cos(2x+\frac{\pi}{3})\cos\frac{\pi}{3}-\cos(2x+\frac{\pi}{3})]$

$\therefore$                     $f(x)=\frac{5}{4}.$      $\forall$   $x$.

Hence ,     $f(\frac{\pi}{8})=\frac{5}{4}$   and   therefore   $gof(\frac{\pi}{8})=g(\frac{5}{4})=1$

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