3

I need to prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$. I have already checked that this is correct for $n=1$ and $n=2$, but I'm not able to prove this identity by induction. Now I was thinking of making a shift to the imaginary numbers by saying : $\sin(nx) + \sin((n+2) x) = Im\{e^{i n x}(1 + e^{2ix})$, but I have no clue how to continue.

vadim123
  • 82,796
user54297
  • 1,036

2 Answers2

4

Recall the formula $$\sin(A) + \sin(B) = 2 \cos((B-A)/2) \sin((A+B)/2)$$

  • It is that simple?? what is the name of this identity? Thanks – user54297 May 06 '13 at 19:55
  • I am not sure if there is any name associated with this identity. You may try googling for sum of sines formula. –  May 06 '13 at 19:58
  • Prosthaphaeresis formulas. Or Simpson's formulas, most of the time : http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html – mwoua May 06 '13 at 20:30
  • Many tables nowadays list these as the "sum-to-product" formulas. (Prosthaphaeresis is what people used before the introduction of logarithms, if you can imagine...) – colormegone May 07 '13 at 02:18
1

Use the addition theorem $$\sin(\alpha+\beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta$$ This implies $$\sin(\alpha-\beta) + \sin(\alpha+\beta) = 2\sin\alpha \cos\beta$$ Now set $\alpha = (n+1)x$, $\beta = x$.

ccorn
  • 9,803