Explanation

(a)
Since, $\cos{\theta = \frac{8}{17}}$ and $0 < \theta < \frac{\pi}{2}$
$\Rightarrow \sin{\theta = \sqrt{1 - \frac{8^{2}}{17^{2}}} = \frac{15}{17}}$
Now, $\cos{(30{^\circ} + \theta) + \cos{(45{^\circ} - \theta) + \cos{(120{^\circ} - \theta)}}}$
$= \cos{30{^\circ}\cos{\theta - \sin{30{^\circ}\sin{\theta + \cos{45{^\circ}\cos{\theta + \sin{45{^\circ}\sin{\theta + \cos{120{^\circ}\cos{\theta + \sin{120{^\circ}\sin\theta}}}}}}}}}}}$
$= \cos{\theta\left( \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} - \frac{1}{2} \right) - \sin{\theta\left( \frac{1}{2} - \frac{1}{\sqrt{2}} - \frac{\sqrt{3}}{2}\ \right)}}$
$= \frac{8}{17}\left( \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} - \frac{1}{2} \right) + \frac{15}{17}\left( \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} - \frac{1}{2} \right)$
$= \frac{23}{17}\left( \frac{\sqrt{3} - 1}{2} + \frac{1}{\sqrt{2}} \right)$