3.33: AP linked to GP; trig GP infinity sum

An AP has first term $9$, last term $159$ and sum $2604$. The $12$th AP term is the first GP term; the $8$th AP term is the second GP term.

Find the sum of the first $6$ terms of the GP.

A GP has first term $\sin\theta$ and ratio $\cos\theta$ for $45^{\circ}\leqslant\theta\leqslant135^{\circ}$. Show $S_{\infty}$ exists.

Show $S_{\infty}=\operatorname{cosec}\theta+\cot\theta$.