Explanation

(a)
$\cos^{2}{\frac{\pi}{16} + \cos^{2}{\frac{3\pi}{16} + \cos^{2}{\frac{5\pi}{16} + \cos^{2}\frac{7\pi}{16}}}}$
$= \cos^{2}{\frac{\pi}{16} + \cos^{2}{\frac{3\pi}{16} + \cos^{2}{\left( \frac{\pi}{2} - \frac{3\pi}{16} \right) + \cos^{2}\left( \frac{\pi}{2} - \frac{\pi}{16} \right)}}}$
$= \cos^{2}{\frac{\pi}{16} + \cos^{2}{\frac{3\pi}{16} + \sin^{2}{\frac{3\pi}{16} + \sin^{2}{\frac{\pi}{16} = 1 + 1 = 2}}}}$