ACJC/2022/JC1/Promo/Q06

The function $\mathrm{f}$ is such that $\mathrm{f}(r)=\cos r \theta$.
(i) Show that $\mathrm{f}(2(r-1))-\mathrm{f}(2 r)=k \sin [(2 r-1) \theta] \sin \theta$, where $k$ is a constant to be determined.

[2]

(ii) Using your result in (i), show that $\sum_{r=1}^n \sin [(2 r-1) \theta]=\frac{\sin ^2 n \theta}{\sin \theta}$ using the method of difference.

[3]

(iii) Hence find $\sum_{r=1}^n \sin [(2 r+1) \theta]$ in terms of $n$ and $\theta$.

[2]