I'm trying to prove that the functions:
$cot(x), cot(2x), \dots, cot(nx)$ are linearly independent.
My idea was to use mathematical induction, that is:
For $n = 1, \hspace{0.5cm} \alpha_1 cot(x) \equiv 0 \Leftrightarrow \alpha_1 = 0 $
Now suppose this holds for n,
$\alpha_1cot(x) + \alpha_2cot(2x) + \dots + \alpha_ncot(nx) \equiv 0 \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} \alpha_1 = \alpha_2 = \dots = \alpha_n = 0$
We have to prove that
$\alpha_1cot(x) + \alpha_2cot(2x) + \dots + \alpha_ncot(nx) + \alpha_{n+1} cot((n+1)x) \equiv 0 \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} \alpha_1 = \alpha_2 = \dots = \alpha_n = \alpha_{n+1} = 0 \hspace{0.4cm} \dots \dots (1)$
I tried to differentiate both sides and we get:
$$\sum_{k=1}^n -\alpha_k(1 + (cot(kx))^2) \equiv 0 \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} \sum_{k=1}^n \alpha_k(1 + (cot(kx))^2) \equiv 0 $$
However, now I'm not sure how to proceed. I would appreciate any suggestions for the way of my approach or any other easier way to prove it!