Express the trigonometric identity $\cos nx$ as an infinite series of $\sin^2 x/2$.

Question

The trigonometric identity $\cos nx$ is expressed as an infinite series only in terms of $\sin^2 \frac{x}{2}$ as follows. $$ \cos nx = 1 + \sum_{l=1}^n (-1)^l \frac{2^{2l}}{(2l)!} \prod_{k=0}^{l-1} (n^2 - k^2) \sin^{2l} \frac{x}{2}$$ This is given in the literature but the authors have not provided any proof to this equation. I have tried the proof using the Euler formula and Binomial theorem, but could not succeed. Can anyone please provide the proof for this equation, or at least a guide on how to prove this ?

Thank You.

Answer 1

The Chebyshev polynomials verify $T_n(\cos x) = \cos nx$. This means that $\cos nx$ can be written as a finite sum of powers of $\cos x$. Now simply use the identity $\cos x = 1-2\sin^2(x/2)$ to obtain that $\cos nx$ is a polynomial of $\sin^2(x/2)$.

Answer 2

Too long for a comment.

In the book "Numerical Computation of Internal and External Flows - The Fundamentals of Computational Fluid Dynamics" second edition, by Charles Hirsch [chapter $8$, page $351$], equation $(8.1.46)$ is exactly written as $$\cos (m\phi) = 1 + \sum_{l=1}^m (-)^l \frac{2^{2l}}{(2l)!} \prod_{k=0}^{l-1} (m^2 - k^2) \sin ^{2 l}\left(\frac{\phi }{2}\right)$$ and, as I wrote in a early comment, I have problems with it. As you wrote, I assume that $(-)^l$ stands for $(-1)^l$ which gives the formula in the post.

It works for $m=1$. But let us try it for $m=2$; hoping that I am not mistaken, the rhs write $$1-8 \sin ^2\left(\frac{\phi}{2}\right)+8 \sin ^8\left(\frac{\phi}{2}\right)$$ while using $t=\sin \left(\frac{\phi}{2}\right)$ $$\cos(2\phi)=1-8t^2+8t^4$$

In short, I do not understand.

Edit

Written as $$\cos (m\phi) = 1 + \sum_{l=1}^m (-1)^l \frac{2^{2l}}{(2l)!}\sin ^{2 l}\left(\frac{\phi }{2}\right) \prod_{k=0}^{l-1} (m^2 - k^2) $$ Using Pochhammer symbols $$\prod_{k=0}^{l-1} (m^2 - k^2)=-(-1)^l m^2 (1-m)_{l-1} (m+1)_{l-1}$$ and the summation is $$\cos \left(2 m \sin ^{-1}\left(\sin \left(\frac{\phi }{2}\right)\right)\right)-1$$