proof of $\sin(x)$ infinite series I can't seem to find a proof for this
$$\sin( x )= x - \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!}+\cdots$$
Which euler used to proof the basel problem. 
Would be great if it anyone show us how to this equality
Thanks

Assume $\sin x=a_0+a_1 x^1+a_2 x^2 + a_3 x^3... $
Note that $\sin 0=0\implies a_0=0$
Now differentiate both sides.
$\cos x = a_1+2a_2x+3a_3x^2...$
$\cos 0 =1 \implies a_1=1$
Differentiate again
$-\sin x=2a_2+3!a_3x...$
$\sin 0=0 \implies a_2=0$
Differentiate again
$-\cos x=3! a_3...$
$\cos 0 =1 \implies a_3=-\frac{1}{3!}$
Hence you can find the pattern.
