I'm looking at a solution for a problem and one of the steps says that:
$\sin(100\pi t)\sin(500\pi t) = \frac{1}{2}[\sin(100\pi+500\pi)t-\sin(500\pi-100\pi)t]$
The thing is, I don't recognize that identity and can't find it searching online. Is it valid, or did the person who wrote the solution make a mistake?
As a side note, in the next step they somehow combine that to get:
$\frac{1}{2}\sin(200\pi)t$
Which I don't see how they did either.
First the identity is the next: $$\sin \theta \sin \varphi = {{\cos(\theta - \varphi) - \cos(\theta + \varphi)} \over 2}$$ Then applying it: $$\sin(100\pi t)\sin(500\pi t) = \frac{[\cos(100\pi-500\pi)t-\cos(100\pi+500\pi)t]}{2}=\frac{[\cos(-400\pi)t-\cos(600\pi)t]}{2}$$ From that point I don't see how is possible to reduce it.