Explanation

(d)
Let $I = \int_{0}^{\frac{\pi}{8}}{\cos^{3}{4\theta d\theta}}$
$= \int_{0}^{\frac{\pi}{8}}\left( \frac{3\cos{4\theta} + \cos{12\theta}}{4} \right)d\theta$
$= \frac{1}{4}\left\lbrack \frac{3\sin{4\theta}}{4} + \frac{\sin{12\theta}}{12} \right\rbrack_{0}^{\frac{\pi}{8}}$
$= \frac{1}{4}\left\lbrack \frac{3}{4}\left( \sin\frac{\pi}{2} - 0 \right) + \frac{1}{12}\left( \sin\frac{3\pi}{2} - 0 \right) \right\rbrack$
$= \frac{1}{4}\left\lbrack \frac{3}{4} + \frac{1}{12}( - 1) \right\rbrack = \frac{1}{4}\left( \frac{8}{12} \right) = \frac{1}{6}$