Explanation

(a)
Given, $f(x) = \sin^{2}{x + \sin^{2}\left( x + \frac{\pi}{3} \right)} + \cos{x\cos\left( x + \frac{\pi}{3} \right)}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sin^{2}{x + \left\lbrack \sin{x\cos\frac{\pi}{3}} + \cos{x\sin\frac{\pi}{3}} \right\rbrack^{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \cos{x\left\lbrack \cos{x\cos{\frac{\pi}{3} - \sin{x\sin\frac{\pi}{3}}}} \right\rbrack}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sin^{2}{x + \left\lbrack \frac{\sin x}{2} + \cos{x.\frac{\sqrt{3}}{2}} \right\rbrack}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \cos{x\left\lbrack \frac{\cos x}{2} - \sin{x.\frac{\sqrt{3}}{2}} \right\rbrack}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sin^{2}{x + \frac{\sin^{2}x}{4} + \frac{3\cos^{2}x}{4} + \sin{x\cos{x.\ \frac{\sqrt{3}}{2}}}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \frac{\cos^{2}x}{2} - \sin{x\cos\frac{\sqrt{3}}{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{5\sin^{2}x}{4} + 5\frac{\cos^{2}x}{4} = \frac{5}{4}$
$\therefore\ \ \ gof(x) = g\left\lbrack f(x) \right\rbrack = g\left( \frac{5}{4} \right) = 1$ (given)