(i)[7]
Demonstrate that $\tan^2 x + \cos^2 x = \sec^2 x + \tfrac{1}{2}\cos 2x - \tfrac{1}{2}$ and hence determine the exact value of $\int_{0}^{\frac{1}{4}\pi} (\tan^2 x + \cos^2 x) \, dx$.
(ii)[4]
The region bounded by the curve $y = \tan x + \cos x$ and the lines $x = 0$, $x = \tfrac{1}{4}\pi$ and $y = 0$ appears in the diagram. Find the exact volume of the solid formed when this region is rotated completely about the $x$-axis.