Let $\mathrm{f}(r)=\frac{\sin \left[ (2r+1)\theta \right]}{\cos \theta }$, where $r$ i

Question

Let $\mathrm{f}(r)=\frac{\sin \left[ (2r+1)\theta \right]}{\cos \theta }$, where $r$ is positive integer. Show that $\mathrm{f}(r)-\mathrm{f}(r-1)=A\cos \left( Br\theta \right)\tan \left( \theta \right)$, where $A$ and $B$ are to be determined. By using the result in part (a), show that $\cos 2\theta +\cos 4\theta +...+\cos 2N\theta =\frac{1}{2}\left[ \frac{\sin \left[ \left( 2N+1 \right)\theta \right]}{\sin \theta }-1 \right]$.

Accepted Answer

$$\cos 2	heta + \cos 4	heta + \cdots + \cos 2N	heta = \dfrac{1}{2}\left[\dfrac{\sin(2N+1)	heta}{\sin	heta} - 1ight]$$

CJC Sigma Notations Tutorial Q4

CJC Sigma Notations Tutorial Q4

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