Archimedes' Limits to Value of Pi/Lemma 1

Lemma for Archimedes' Limits to Value of Pi

$\ds \cot \dfrac \phi 2 = \cot \phi + \csc \phi$

$\ds \tan \dfrac \phi 2$

$=$

$\ds \paren {\csc \phi - \cot \phi}$

$\ds \leadsto \ \ $

$\ds \cot \dfrac \phi 2$

$=$

$\ds \dfrac 1 {\paren {\csc \phi - \cot \phi} }$

$\ds $

$=$

$\ds \dfrac {\csc \phi + \cot \phi} {\paren {\csc \phi - \cot \phi} \paren {\csc \phi + \cot \phi} }$

multiplying top and bottom by $\csc \phi + \cot \phi$

$\ds $

$=$

$\ds \dfrac {\csc \phi + \cot \phi} {\paren {\csc^2 \phi - \cot^2 \phi} }$

$\ds \cot \dfrac \phi 2$

$=$

$\ds \cot \phi + \csc \phi$

$\Box$