<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
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\begin{document}

{\bf Question}

Show that $x \mapsto \cos x$ has a unique attracting fixed point,
and no other periodic points.  What about $x \mapsto \sin x$?
\medskip

{\bf Answer}

Graph of $y=\cos x$ meets the diagonal $y=x$ at one point; we have
$\left|\ds\frac{d}{dx} \cos x \right|=|\sin x|&lt;1$ when $x \ne
\ds\frac{n\pi}{2}$ so the fixed point is attracting.  Graph of $y=
\sin x$ meets the diagonal only at $x=0$, with $|\sin(x)|&lt;|x|$ for
nonzero x: thus 0 is attracting.(Compare to $x-x^3$)
\end{document}
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