Sum of sinx+sin3x+sin5x+......sin(2n-1)x

Question

Options are

1. n/2 cosx- 1sin(nx)/2sinx . cos(n+2)x
2. n/2.sinx-1/2sin(nx)
3. n/2.cosx - cos(n+2)x
4. sinx +sin(nx)

Answer 1

Hint

let $z=\cos\theta+i\sin\theta$, use $$\operatorname{Im} \{ 1+z+z^2+...+z^{2n}\}=\operatorname{Im}\left(\frac{1-z^{2n+1}}{1-z}\right)$$ and $$\operatorname{Im} \{1+z^2+z^4+...+z^{2n}\}=\operatorname{Im}\left(\frac{1-z^{2n+2}}{1-z^2}\right)$$ $$\sin\theta+\sin3\theta+\sin5\theta+...+\sin(2n-1)\theta=\operatorname{Im}\left(\frac{1-z^{2n+1}}{1-z}\right)-\operatorname{Im}\left(\frac{1-z^{2n+2}}{1-z^2}\right)$$