- 4

- 0

**1. Homework Statement**

Prove that [itex]\displaystyle \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N}[/itex]

**2. The attempt at a solution**

Here's my attempt using induction:

Let [itex]P(n) [/itex] be the statement given by

[tex] \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N}[/tex]

P(1):

[tex] \int_{0}^{\pi}\frac{1-\cos(x)}{1-\cos(x)} dx=\int_{0}^{\pi}dx=\pi[/tex]

[tex](1)\pi=\pi[/tex]

**P(1) holds true.**

Let [itex]P(n)[/itex] be true.

Now, we need to show that [itex]P(n+1)[/itex] is true.

[tex]\int_{0}^{\pi} \frac{1-\cos{x(n+1)}}{1-\cos(x)}dx=\int_{0}^{\pi}\frac{1-[\cos(nx)\cos(x)-\sin(nx)\sin(x)]}{1-\cos(x)} dx[/tex]

I don't know how to proceed from here.

Furthermore, I would like to know if I could prove the statement by solving the integral.