Can someone help me prove the following by mathematical induction:
$$\sin((2n+1)x)=\sin(x)(1+2 \sum_{k=1}^{n} \cos(2kx))$$
I was told to use induction on $n$; however I keep getting stuck. Any help would be greatly appreciated!
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If $n=0$, this is trivially true. Suppose $n-1$ is true, let's consider $n$, it suffices to show $$\sin((2n+1)x)-\sin((2n-1)x)=2\sin(x)\cos(2nx)$$ now do you see how to finish?
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Yes! Thank you so much for your help!! – user75244 Apr 30 '13 at 19:05
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1The formula is $\sin(x)-\sin(y)=\color{#C00000}{2}\cos\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right)$ – robjohn Apr 30 '13 at 19:06