2009/011) Prove that (sin x)/x = cos(x/2)*cos(x/4)*cos(x/8)*cos(x/16)..

Proof:
Sin x = 2 cos (x/2) sin (x/2)
= 2 cos (x/2) (2 cos(x/4) sin (x/4))
= 2^n cos (x/2) cos(x/4) cos(x/8)cos(x/16) …. cos (x/2^n) sin (x/2^n)

So (sin x)/x = 2^n cos (x/2) cos(x/4) cos(x/8)cos(x/16) …. cos (x/2^n) sin (x/2^n)/ x
= cos (x/2) cos(x/4) cos(x/8)cos(x/16) …. cos (x/2^n) sin (x/2^n)/ (x/2^n)
as n goes to infinite sin (x/2^n)/ (x/2^n) goes to 1 and we get

(sin x)/x = cos (x/2) cos(x/4) cos(x/8)cos(x/16) ….


Proved