User:Ybab321/Sandbox

Feel free to help or even complete whatever proofs appear here.

= Cosine Multiple Angle Formula =

Theorem

 * $\displaystyle \cos n \theta = n \sum^{\left\lfloor{\frac n 2}\right\rfloor}_{k \mathop = 0} \frac{\left({-1}\right)^k \left({n-k-1}\right)! 2^{n-2k-1} \cos^{n-2k} \theta}{k!\left({n-2k}\right)!}$

Proof
The proof proceeds by induction on $n$.

Basis for the Induction
The case $n = 1$ is trivial

The case $n = 2$ is verified as follows:

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge 1$.

Assume $\displaystyle \cos n \theta = n \sum^{\left\lfloor{\frac n 2}\right\rfloor}_{k \mathop = 0} \frac{\left({-1}\right)^k \left({n-k-1}\right)! 2^{n-2k-1} \cos^{n-2k} \theta}{k!\left({n-2k}\right)!}$ holds for $n$.

This is our induction hypothesis.

Induction Step
This is our induction step: