hello I am given prove that

$\displaystyle \frac{sin\theta}{2} + \frac{sin2\theta}{2^2} + \frac{sin3\theta}{2^3} +...........= \frac{2sin\theta}{5-4cos\theta} $

what I have done

this is a Geometric progression therefore sum to infinity = $\displaystyle \frac{a}{1-r} $

a= $\displaystyle \frac{sin\theta}{2} $

r = $\displaystyle \frac{sin2\theta}{2^2} / \frac{sin\theta}{2} $

r = $\displaystyle \frac{sin2\theta}{2sin\theta} $

substituting back into the equation I get

sum to infinity = $\displaystyle \frac{2sin^2\theta}{2-2cos\theta} $

can someone please show me where I am going wrong, thanks.