2

I refer to these formulas:

$\sin(x±y)$ and $\cos(x±y)$

Is there an obvious or intuitive proof to derive their simpler identities in terms of $\sin(x), \sin(y), \cos(x), \cos(y)$?

I am tagging this with complex analysis because I'm open to such an explanation if it's simple enough to understand.

Blue
  • 75,673
user525966
  • 5,631

1 Answers1

2

If you know complex analysis, it follows from $\exp(x+y)=\exp(x)\exp(y)$, by using $\exp(ix) = \cos x + i \sin x$ and comparing real and imaginary parts.

Bart Michels
  • 26,355
  • Hm maybe I need to go learn the basics of complex analysis, as I don't see how any of this follows. Why does $\exp(ix) = \sin(x) + i\cos(x)$? – user525966 Feb 21 '18 at 06:53
  • I'm not saying the proof would be elementary. It all depends on which definitions you take. That identity ("Euler's formula") is not hard to see if you define exp, sin and cos using their series expansion, but then $\exp(x+y)=\exp(x)\exp(y)$ is less natural. In fact, Euler's formula gives one possible definition of the complex exponential. They all follow immediately if you define exp, sin, cos as solutions to differential equations, which is perhaps the cleanest way. – Bart Michels Feb 21 '18 at 07:23
  • 3
    It's $\cos x+i\sin x$. – J.G. Feb 21 '18 at 07:37