I'm afraid the equation \cos (\pi A) = E - 2A, \tag{1} where A is an orthogonal matrix, cannot hold, at least in the stated generality. To see a counterexample, consider the matrix J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \tag{2} ...

Writing \cos(\pi t) in terms of \cos(\frac{\pi t}{2^n}) ... the same thing as writing \cos(2^n\theta) in terms of \cos(\theta) ... is a Chebyshev polynomial

Gió May 5, 2015 You can consider it visually as a graph and remember the periodicity of your \displaystyle{\cos} , meaning that it repeats itself at regular intervals (oscillating ...

1 Explanation: Unit circle --> \displaystyle{\cos{{\left({5}\pi\right)}}}={\cos{{\left(\pi+{4}\pi\right)}}}={\cos{\pi}}=-{1}

\displaystyle{1} Explanation: Notice that one complete rotation is \displaystyle{2}\pi : So \displaystyle{8}\pi is just 4 complete rotations. We therefore only need to find: \displaystyle{\cos{{\left({0}\right)}}} ...

The cosine of a integer times \pi is always \pm 1. It happens that when k is even, \cos{k\pi} = 1, when k is odd, \cos{k\pi}=-1 and therefor (-1)^k. The sine is always zero for every k. So ...