Question

If f(x) = sin x, show that f (2x)= 2f(x) f(1/2 π-x).

Answer 1

If $f(x)=sinx,$ then

$f(2x)=sin2x=2sinxcosx$ (double angle formula).

Let's consider the right side of the equality:

$2f(x)f(\frac{\pi}{2}-x)=2sinxsin(\frac{\pi}{2}-x)=2sinxcos x,$

because $sin(\frac{\pi}{2}-x)=cosx$ (co-function identity).

So we have:

$f(2x)=2sinxcosx=2f(x)f(\frac{\pi}{2}-x),$

and the statement is proved.