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I noticed that $\tan(nx)$ can be written as the ratio $P(\tan x) / Q(\tan x)$, where $P$ and $Q$ are polynomials.

Indeed, I found a short proof here Expansion of $\tan(nx)$ in powers of $\tan(x)$

What else can we say? If we do the same idea for $\cos(nx)$ we get the Chebyshev polyomials, and we know many things for those.

a ) So I was wondering if $P(X)/Q(X)$ satisfies any extremality properties, like the Chebyshev polyomials?

b)Or are they useful in Pade approximation? Just guessing.

c)Or they can be gained as a solution of differential equations? with coefficients gotten by recursion relations?

I reached the Chebyshev polyomials in my progress but I am now stuck.

leonbloy
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Dave86
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  • Just a thought (and not a justifiable answer hence why only a comment) but if you consider expansions of functions using Chebyshev polynomials, the derivatives are generally simpler than if you used $\tan(nx)$, For example, $\cos(nx)$ and $\sin(nx)$ just interchange under differentiation, up to constants, whereas expressions containing linear combinations of $\tan(nx)$ involve $\sec^2(nx)$. Basically, why choose a basis that creates complexity with no other obvious benefit? – Paul Sep 29 '18 at 20:17
  • Since your talking about $P(x)/Q(x)$ instead of $P(\tan x)/Q(\tan x)$ in part a, that is taking solutions $x_0$ of $x=\tan x$. $\tan n x_0 = \frac{P(x_0)}{Q(x_0)}$ – AHusain Sep 29 '18 at 20:21

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