trigonometry - Evaluate the sum of the infinite series $1+cos x + cos^2 x + cos ^3 x ...$ for $0

Evaluate the sum of the infinite series $1+\cos x + \cos^2 x + \cos ^3 x ...$ for $0

So am I correct in thinking that $$1+\cos x + \cos^2 x + \cos ^3 x ...=\sum ^\infty _{n=0} \cos^n x$$ which is just a geometric series with common ratio $\cos x$ and first term 1. So the sum of the series should be $$\sum ^\infty _{n=0} \cos^n x =\frac{1}{1-\cos x}$$ However, the answer to the question is $\frac1 2 \csc^2(\frac x 2)$. Is my method not correct or do I need to apply some identities, if so how do I get it into this form? any help would be great.

Answer

What you did is fine. And then you use the fact that\begin{align}1-\cos(x)&=1-\left(\cos^2\left(\frac x2\right)-\sin^2\left(\frac x2\right)\right)\\&=2\sin^2\left(\frac x2\right).\end{align}