We have to find
g(x)=cosx+cos3x+cos5x+⋯+cos(2n−1)x
I could not get any good idea .
Intialy I thought of using
cosa+cosb=2cos(a+b)/2cos(a−b)/2
Answer
Let z=cosθ+isinθ i.e. z=eiθ
Your sum:eiθ+e3iθ+e5iθ+...e(2n−1)iθ
This is a GP with common ratio e2iθ
Therefore sum is a(rn−1)r−1
eiθ(e2niθ−1)e2iθ−1
(cosθ+isinθ)(cos(2nθ)+isinθ−1)cos(2θ)+isin(2θ)−1
Computing it's real part should give you the answer
Acknowledgement:Due credits to @LordShark Idea
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