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Question Number 29848 by abdo imad last updated on 12/Feb/18

$$find\sum _{k=0}^{n}cos\left(kx\right)and\sum _{k=0}^{n}sin\left(kx\right).$$

Commented by Tinkutara last updated on 13/Feb/18

$$1+\cos x+\cos 2x+...+\cos nx$$$$=\frac{\sin \frac{(n+1)x}{2}}{\sin \frac{x}{2}}\cos \left(\frac{nx}{2}\right)$$$$\sin x+\sin 2x+\sin 3x+...+\sin nx$$$$=\frac{\sin \frac{nx}{2}}{\sin \frac{x}{2}}\sin (x+\frac{(n-1)x}{2})$$$$=\frac{\sin \frac{nx}{2}}{\sin \frac{x}{2}}\sin \left(\frac{(n+1)x}{2}\right)$$

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