Show that $\sin \left( n+1 \right)-\sin \left( n-1 \right)=k\cos n$, where $k$ is an exact constant

Question

Show that $\sin \left( n+1 \right)-\sin \left( n-1 \right)=k\cos n$, where $k$ is an exact constant to be determined. [2] Hence or otherwise, find $\cos \left( 1 \right)+\cos \left( 2 \right)+\cos \left( 3 \right)+...+\cos N$ in terms of $N$, expressing your answer in the form $a\left( \sin \left( N+1 \right)+\sin N \right)+b$, where $a$ and $b$ are constants to be determined. [4]

Accepted Answer

$k=2\sin1$; $\displaystyle \sum_{n=1}^{N}\cos n=\frac{\sin N+\sin(N+1)}{2\sin1}-\frac12$

2019 RVHS Promo Q2

2019 RVHS Promo Q2

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