In the course of working on some correlation problem, I happened to notice that in the vicinity of $0$, $\cos^n x$ is a pretty darned good approximation to $\exp\left(\frac{-nx^2}{2}\right)$. In other words, one can approximate a normal distribution with a high power of cosine.

This turned out to be useful to me in the opposite direction, because I had a PDF that was a high power of cosine, and I could use the normal approximation. But it occurred to me that this is simple enough to see (I arrived at it using the first two terms of the Taylor series for cosine, but there are other ways, I imagine) that it must have been noticed before, but not so old that it would be prehistoric (mathematically speaking), and without a known discoverer.

So, does anyone know who first noticed this? Or is this lost to the sands of time because it is so easily found?

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    $\begingroup$ $$\cos^n(x)=e^{n\log(\cos(x))}\approx e^{-n x^2/2}$$ if $n$ is large enough. This kind of approximation plays a big role in the approixmation of a class of integrals in the large $n$ limit invented by Laplace, so my bet is on him $\endgroup$ – tired Apr 21 '17 at 16:46
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    $\begingroup$ Indeed it is easy to find. One just needs to notice $\tan(x)\geq x$ on $(0,\pi/2)$, integrate both sides, exponentiate. $\endgroup$ – Jack D'Aurizio Apr 21 '17 at 16:49
  • $\begingroup$ The approximation error can be deduced from the inequality $$ n b^{n-1}(a-b)\leq a^n-b^n \leq n(a-b)a^{n-1} $$ (with the assumption $a>b>0$) $\endgroup$ – Jack D'Aurizio Apr 21 '17 at 16:50
  • $\begingroup$ @JackD'Aurizio: Thanks! However, I don't have any problem seeing the inequality, nor am I looking in this question for the other ways of arriving at it. I'm primarily curious about the history of it (if any). $\endgroup$ – Brian Tung Apr 21 '17 at 17:08
  • $\begingroup$ @tired: That sounds plausible. Maybe someone will come up with some kind of citation. $\endgroup$ – Brian Tung Apr 21 '17 at 17:09

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