Problem

The first and second terms of an arithmetic progression are $\tan\theta$ and $\sin\theta$ respectively, where $0 < \theta < \frac{1}{2}\pi$.

(a) Given that $\theta = \frac{1}{4}\pi$, find the exact sum of the first $40$ terms of the progression.

The first and second terms of a geometric progression are $\tan\theta$ and $\sin\theta$ respectively, where $0 < \theta < \frac{1}{2}\pi$.

(b) (i) Find the sum to infinity of the progression in terms of $\theta$.

(ii) Given that $\theta = \frac{1}{3}\pi$, find the sum of the first $10$ terms of the progression. Give your answer correct to $3$ significant figures.