Trigonometry involving usage of series

Question

What is the value of $[\cos(\pi)/2n + 1] + [\cos(3\pi)/2n + 1] + \cos(5\pi)/2n + 1]$ ..... upto $'n'$ terms equals?

Can anyone, please solve this question for me?

Answer 1

Using the fact that $\cos(k\pi)=-1$, for all odd values of $k$, the summation reduces to $$ [\frac{-1}{2n}+1]+[\frac{-1}{2n}+1]+... \text{upto 'n' terms}$$ which equals $$ n(\frac{-1}{2n}+1) = -\frac{1}{2}+n$$