Prove that for all $n \in N$ $\cos \alpha+ \cos ( \alpha + \beta ) + \cos ( \alpha + 2 \beta )+...+ \cos ( \alpha + (n-1) \beta )$ $= \large\frac{ \cos \bigg( \alpha + \bigg( \Large\frac{n-1}{2} \bigg) \beta \bigg) \sin \bigg( \Large\frac{n \beta}{2} \bigg)}{\sin \Large\frac{\beta}{2}}$ by using principle of mathematical induction.

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