$$help \enspace me \enspace ummah \enspace forum \enspace :D$$ \newline If $$C = 1+cos\theta+...+cos(n-1)0,$$ $$S = sin\theta+...+sin(n-1)0,$$ prove that $$C=\frac{sin\frac{n^\theta}{2}}{sin\frac{\theta}{2}} cos\frac{(n-1)\theta}{2} || S = \frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}}sin\frac{(n-1)\theta}{2}$$ if $$\theta \neq 2k\pi, k\in \mathbb{Z}$$ Attempt $$C+iS = 1+(cos\theta+isin\theta)+...+(cos(n-1)\theta+isin(n-1)\theta)$$ $$=1+e^{i\theta}+...+e^{i(n-1)\theta}$$ $$=1+z+...+z^{n-1}, \enspace where \enspace z=e^{i\theta}$$ $$=\frac{1-z^n}{1-z}, \enspace if \enspace z\neq1$$ $$=\frac{1-e^{in\theta}}{1-e^{i\theta}}=\frac{e^\frac{in\theta}{2}(e^\frac{-in\theta}{2})-e^\frac{in\theta}{2}}{e\frac{i\theta}{2}(e\frac{-i\theta}{2})-e\frac{i\theta}{2}}$$