Math Problem 6

(a) Prove, by mathematical induction, that

$\displaystyle \cos x + \cos 2x + \cdots + \cos(nx) = \frac{\sin\left(\left(n+\tfrac12\right)x\right)}{2\sin\left(\tfrac{x}{2}\right)} -\frac12$

for all positive integers $n$, where $\sin\left(\tfrac{x}{2}\right) \neq 0$.

(b) Hence, or otherwise, evaluate

$\displaystyle \lim_{x \to 0}\frac{\sin\left(\left(n+\tfrac12\right)x\right)}{\sin\left(\tfrac{x}{2}\right)}$ , where $n$ is a positive integer.

($6 + 2$ marks)

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