Product of multiple sines Is there an identity for the following equation:
$\ f(x) = \sin(x.\pi/2)\cdot \sin(x.\pi/3)\cdot \sin(x.\pi/4) $
I am looking for an equation similar to this equation
 A: Express everything in terms of the trigonometric functions of $\frac {\pi x}{12}$ and working a little
$$4\sin \left(\frac{\pi  x}{2}\right) \sin \left(\frac{\pi  x}{3}\right) \sin
   \left(\frac{\pi  x}{4}\right)=\sin \left(\frac{\pi  x}{12}\right)+\sin \left(\frac{5 \pi    x}{12}\right)+\sin \left(\frac{7 \pi  x}{12}\right)-\sin \left(\frac{13 \pi 
   x}{12}\right)$$ It would be much less funny do add the next term; the common factor being $\frac {\pi x}{60}$, we should have
$$8\sin \left(\frac{\pi  x}{2}\right) \sin \left(\frac{\pi  x}{3}\right) \sin
   \left(\frac{\pi  x}{4}\right)\left(\frac{\pi  x}{5}\right)=\cos \left(\frac{7 \pi  x}{60}\right)+\cos \left(\frac{13 \pi  x}{60}\right)-$$ $$\cos
   \left(\frac{17 \pi  x}{60}\right)+\cos \left(\frac{23 \pi  x}{60}\right)-\cos
   \left(\frac{37 \pi  x}{60}\right)-\cos \left(\frac{47 \pi  x}{60}\right)-\cos
   \left(\frac{53 \pi  x}{60}\right)+\cos \left(\frac{77 \pi  x}{60}\right)$$
All the trick is to expand $\sin(nt)$ in terms of powers of $\sin(t)$ and $\cos(t)$ and then the power reduction formulae (look here).
