<s>[INST] Complex number trigonometry problem Use $cos (n\theta)$ = $\frac{z^n +z^{-n}}{2}$ to express 
$\cos \theta + \cos 3\theta + \cos5\theta + ... + \cos(2n-1)\theta$ as a geometric series in terms of z. Hence find this sum in terms of $\theta$.
I've tried everything in the world and still can't match that of the final answer. Could I pleas have a slight hint on the right path to follow.
Thanks
 [/INST] $ cosθ+cos3θ+cos5θ+\cdot \cdot \cdot+cos(2n−1)θ$
=$ \dfrac{z+z^{-1}}{2} + \dfrac{z^3+z^{-3}}{2} + \dfrac{z^5+z^{-5}}{2} + \cdot \cdot \cdot + \dfrac{z^{(2n+1)}+z^{-(2n-1)}}{2} $ $$\\$$ = $ 2^{-1}(z+z^3+z^5+z^7+\cdot \cdot \cdot + z^{2n-1}) + 2^{-1}( z^{-1}+z^{-3}+z^{-5} +\cdot \cdot \cdot + z^{-(2n-1)}) $
Then apply the formula of geometric series . 
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