Given that $\mathrm{f}\left( r \right)=\sin \left( 2r+1 \right)\theta $, show that $\mathrm{f}\left( r \right)-\mathrm{f}\left( r-1 \right)=A\cos Br\theta \sin \theta $, where $A$ and $B$ are constants to be found.[2]
Use the result above to show that
$\cos 2\theta +\cos 4\theta +...+\cos 2N\theta =\frac{\sin \left( 2N+1 \right)\theta }{2\sin \theta }-\frac{1}{2}$.[3]
Hence find${{\cos }^{2}}2\theta +{{\cos }^{2}}4\theta +...+{{\cos }^{2}}2N\theta $.[3]

Given that $\mathrm{f}\left( r \right)=\sin \left( 2r+1 \right)\theta $, show that $\mathrm{f}\left( r \right)-\mathrm{f}\left( r-1 \right)=A\cos Br\theta \sin \theta $, where $A$ and $B$ are constants to be found.

$\cos 2\theta +\cos 4\theta +...+\cos 2N\theta =\frac{\sin \left( 2N+1 \right)\theta }{2\sin \theta }-\frac{1}{2}$.

Hence find${{\cos }^{2}}2\theta +{{\cos }^{2}}4\theta +...+{{\cos }^{2}}2N\theta $.

2017 RI Promo Q6

2017 RI Promo Q6

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