\magnification=1100
\overfullrule0pt

\input amssym.def
\input prepictex
\input pictex
\input postpictex


% ********************* Definitions ************************************

%\def\widetilde{\mathaccent"0365 }

\def\CC{{\Bbb C}}
\def\FF{{\Bbb F}}
\def\HH{{\Bbb H}}
\def\NN{{\Bbb N}}
\def\OO{{\Bbb O}}
\def\QQ{{\Bbb Q}}
\def\RR{{\Bbb R}}
\def\ZZ{{\Bbb Z}}

\def\cA{{\cal A}}

\def\cB{{\cal B}}
\def\cR{{\cal R}}
\def\cZ{{\cal Z}}
\def\cC{{\cal C}}
\def\cR{{\cal R}}

\def\fb{\frak{b}}
\def\fg{\frak{g}}
\def\fh{\frak{h}}
\def\fn{\frak{n}}

\def\Card{\hbox{Card}}
\def\End{\hbox{End}}
\def\Hom{\hbox{Hom}}
\def\Ind{\hbox{Ind}}
\def\Id{\hbox{Id}}
\def\codim{\hbox{codim}}

\def\diag{\hbox{diag}}
\def\id{\hbox{id}}
\def\im{\hbox{im}}
\def\tr{\hbox{tr}}
\def\Tr{\hbox{Tr}}

\def\sech{\hbox{sech}}
\def\csch{\hbox{csch}}
\def\coth{\hbox{coth}}

\def\fbox{}


% ********************* FONTS ************************************

\font\smallcaps=cmcsc10
\font\titlefont=cmr10 scaled \magstep1
\font\titlefontbold=cmbx10 scaled \magstep1
\font\titlesubfont=cmr10 scaled \magstep1
\font\sectionfont=cmbx10
\font\tinyrm=cmr10 at 8pt

% ******************** SECTION HEADERS ***************************

\newcount\sectno
\newcount\subsectno
\newcount\resultno

\def\section #1. #2\par{
\sectno=#1
\resultno=0
\bigskip\noindent{\sectionfont #1.  #2}~\medbreak}

\def\subsection #1\par{\bigskip\noindent{\it  #1} \medbreak}

%******************* MATHEMATICAL LABELS **************************

\def\prop{ \global\advance\resultno by 1
\medskip\noindent{\bf Proposition \the\sectno.\the\resultno. }\sl}
\def\lemma{ \global\advance\resultno by 1
\medskip\noindent{\bf Lemma \the\sectno.\the\resultno. }
\sl}
\def\fact{ \global\advance\resultno by 1
\medskip\noindent{\bf Fact \the\sectno.\the\resultno. }
\sl}

\def\remark{ \global\advance\resultno by 1
\medskip\noindent{\bf Remark \the\sectno.\the\resultno. }}
\def\example{ \global\advance\resultno by 1
\medskip\noindent{\bf Example \the\sectno.\the\resultno. }\sl}
\def\cor{ \global\advance\resultno by 1
\medskip\noindent{\bf Corollary \the\sectno.\the\resultno. }\sl}
\def\thm{ \global\advance\resultno by 1
\medskip\noindent{\bf Theorem \the\sectno.\the\resultno. }\sl}
\def\defn{ \global\advance\resultno by 1
\medskip\noindent{\it Definition \the\sectno.\the\resultno. }\slrm}
\def\endthm{\rm\medskip}
\def\thmend{\rm\medskip}
\def\endlemma{\rm\medskip}
\def\endfact{\rm\medskip}
\def\endexample{\rm\medskip}
\def\endprop{\rm\medskip}
\def\endcor{\rm\medskip}
\def\pf{\rm\smallskip\noindent{\it Proof. }}
\def\endpf{\qed\hfil\medskip}
\def\pfend{\qed\hfil\medskip}
\def\note{\smallbreak\noindent{Note:}}
\def\enddefn{\rm\medskip}

%Homemade Struts:
\newbox\strutAbox
\setbox\strutAbox=\hbox{\vrule height 12pt depth6pt width0pt}
\def\strutA{\relax\copy\strutAbox}
\newbox\strutBbox
\setbox\strutBbox=\hbox{\vrule height 10pt depth5pt width0pt}
\def\strutB{\relax\copy\strutBbox}
\newbox\strutDbox
\setbox\strutDbox=\hbox{\vrule height 11pt depth5pt width0pt}
\def\strutD{\relax\copy\strutDbox}
%high strut:
\newbox\strutHbox
\setbox\strutHbox=\hbox{\vrule height 11pt depth1pt width0pt}
\def\strutH{\relax\copy\strutHbox}
%low strut:
\newbox\strutLbox
\setbox\strutLbox=\hbox{\vrule height 1pt depth5pt width0pt}
\def\strutL{\relax\copy\strutLbox}



% hack to ignore lots of typed stuff....
\def\ignore#1{\relax}

% ******************  QED SIGNS  *********************************

\def\qed{\hbox{\hskip 1pt\vrule width4pt height 6pt depth1.5pt \hskip 1pt}}

\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.2pt}
\hrule height.2pt}}}}
\def\square{\mathchoice\sqr54\sqr54\sqr{3.5}3\sqr{2.5}3}
\def\whiteslug{\bf $ \square $ \rm}  % open square


%*************** EQUATIONS WITH NUMBERS **************

\def\formula{\global\advance\resultno by 1
\eqno{(\the\sectno.\the\resultno)}}
\def\formulano{\global\advance\resultno by 1 (\the\sectno.\the\resultno)}
\def\tableno{\global\advance\resultno by 1
\the\sectno.\the\resultno. }
\def\lformula{\global\advance\resultno by 1
\leqno(\the\sectno.\the\resultno)}

%************Commutative diagrams**********************

\def\mapright#1{\smash{\mathop
        {\longrightarrow}\limits^{#1}}}

\def\mapleftright#1{\smash{\mathop
        {\longleftrightarrow}\limits^{#1}}}


\def\mapsrightto#1{\smash{\mathop
        {\longmapsto}\limits^{#1}}}

\def\mapleft#1{\smash{
   \mathop{\longleftarrow}\limits^{#1}}}

\def\mapdown#1{\Big\downarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\def\lmapdown#1{{\hbox{$\scriptstyle#1$}}
\llap {$\vcenter{\hbox{\Big\downarrow}}$} }

\def\mapup#1{\Big\uparrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapne#1{\Big\nearrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapse#1{
%{$\vcenter{
\hbox{$\scriptstyle#1$}
%$}
\rlap{ $\vcenter{\hbox{$\searrow$}}$ }  }
\def\mapnw#1{\Big\nwarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapsw#1{
%\Big
\swarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}


%********** DATING ******************************************
\def\monthname {\ifcase\month\or January\or February\or March\or April\or
May\or June\or
July\or August\or September\or October\or November\or December\fi}

\newcount\mins  \newcount\hours  \hours=\time \mins=\time
\def\now{\divide\hours by60 \multiply\hours by60 \advance\mins by-\hours
     \divide\hours by60         % NOTE: \divide only gives integer answers.
     \ifnum\hours>12 \advance\hours by-12
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ P.M.\else
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ A.M.\fi}
\def\today {\monthname\ \number\day, \number\year}


%**************** PAGE HEADERS *************************

\nopagenumbers
\def\runningtitle{\smallcaps basic trig identities}
\headline={\ifnum\pageno>1\eoheadline\else\firstheadline\fi}
\def\names{\smallcaps a.\ ram}
%\def\firstheadline{\noindent Preliminary Draft \hfill  \today}
\def\firstheadline{}
\def\eoheadline{\ifodd\pageno\oddheadline\else\evenheadline\fi}
\def\oddheadline{\tenrm\hfil\runningtitle\hfil\folio}
\def\evenheadline{\tenrm\folio\hfil{\names}\hfil}


%**************** TITLE *************************
\vphantom{$ $}  %My kludge to get the first page to move down a bit
\vskip.75truein
\centerline{\titlefont The basic trig identities}
\bigskip
\centerline{\rm Arun Ram}
%${}^\ast$ 
\centerline{Department of Mathematics}
\centerline{University of Wisconsin, Madison}
\centerline{Madison, WI 53706 USA}
\centerline{{\tt ram@math.wisc.edu}}
\medskip
\centerline{Version: \today}

%\footnote{}{\tinyrm 
%${}^\ast$ 
%Research partially supported by the National Security Agency
%and by EPSRC Grant GR K99015 at the Newton Institute for
%Mathematical Sciences.}
%\footnote{}{\tinyrm
%\noindent {Keywords:} exponential functions, trigonometric functions}

\bigskip

%**************** ABSTRACT *************************
%\noindent{\bf Abstract.}


\bigskip\bigskip
Define
$$\eqalign{
e^x &= 1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots, \cr
\sin x &= x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}
+{x^9\over 9!}-{x^{11}\over 11!}+{x^{13}\over 13!} -\cdots, \cr
\cos x &= 1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}
+{x^8\over 8!}-{x^{10}\over 10!}+{x^{12}\over 12!} -\cdots, \cr
}
$$
and
$$\tan x ={\sin x\over \cos x}, \quad
\cot x ={1\over \tan x}, \quad
\sec x ={1\over \cos x}, \quad
\csc x ={1\over \sin x}. 
$$


\bigskip\noindent
{\bf Example:}  Explain why $e^{ix} = \cos x+ i\sin x$, if $i = \sqrt{-1}$.
\bigskip
$$\eqalign{
e^{ix} 
&= 1+ix+{(ix)^2\over 2!}+{(ix)^3\over 3!}+{(ix)^4\over 4!}
+{(ix)^5\over 5!}+{(ix)^6\over 6!}+{(ix)^7\over 7!} +\cdots, \cr
&= 1+ix+{i^2x^2\over 2!}+{i^3x^3\over 3!}+{i^4x^4\over 4!}
+{i^5x^5\over 5!}+{i^6x^6\over 6!}+{i^7x^7\over 7!} +\cdots, \cr
&= 1+ix+{i^2x^2\over 2!}+{i\cdot i^2x^3\over 3!}+{((i^2)^2x^4\over 4!}
+{i\cdot(i^2)^2x^5\over 5!}+{(i^2)^3x^6\over 6!}
+{i\cdot(i^2)^3x^7\over 7!} +\cdots, \cr
&= 1+ix+{(-1)x^2\over 2!}+{i\cdot(-1)x^3\over 3!}+{(-1)^2x^4\over 4!}
+{i\cdot(-1)^2x^5\over 5!}+{(-1)^3x^6\over 6!}
+{i\cdot(-1)^3x^7\over 7!} +\cdots, \cr
&= 1+ix-{x^2\over 2!}-i{x^3\over 3!}+{x^4\over 4!}
+i{x^5\over 5!}-{x^6\over 6!} -i{x^7\over 7!} +\cdots, \cr
&= (1-{x^2\over 2!}+{x^4\over 4!} -{x^6\over 6!}+\cdots)
+i(x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!} +\cdots) \cr
&= \cos x + i\sin x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why $\cos(-x)=\cos x$ and $\sin(-x)=-\sin x$.
\bigskip
$$
\eqalign{
\cos (-x) 
&= 1-{(-x)^2\over 2!}+{(-x)^4\over 4!}-{(-x)^6\over 6!}
+{(-x)^8\over 8!}-{(-x)^{10}\over 10!}+{(-x)^{12}\over 12!} -\cdots, \cr
&= 1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}
+{x^8\over 8!}-{x^{10}\over 10!}+{x^{12}\over 12!} -\cdots, \cr
&=\cos x, \cr
}$$
and
$$\eqalign{
\sin(-x) 
&= (-x)-{(-x)^3\over 3!}+{(-x)^5\over 5!}-{(-x)^7\over 7!}
+{(-x)^9\over 9!}-{(-x)^{11}\over 11!}+{(-x)^{13}\over 13!} -\cdots, \cr
&= -x+{x^3\over 3!}-{x^5\over 5!}+{x^7\over 7!}
-{x^9\over 9!}+{x^{11}\over 11!}-{x^{13}\over 13!} +\cdots, \cr
&= -\left(x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}
+{x^9\over 9!}-{x^{11}\over 11!}+{x^{13}\over 13!} -\cdots\right), \cr
&= -\sin x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why $\cos^2 x+\sin^2 x=1$.
\bigskip
$$\eqalign{
1 &= e^0 = e^{ix+(-ix)} \cr
&= e^{ix} e^{-ix} = e^{ix} e^{i(-x)} \cr
&= (\cos x+i\sin x)(\cos(-x)+i\sin(-x)) \cr
&= (\cos x+i\sin x)(\cos x+i(-\sin x)) \cr
&= \cos^2 x - i\sin x\cos x + i \sin x\cos x-i^2\sin^2x \cr
&= \cos^2 x-(-1)\sin^2 x \cr
&= \cos^2 x+\sin^2 x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why\qquad
$\displaystyle{
\eqalign{
\cr
\cos(x+y)&=\cos x\cos y - \sin x\sin y,
\qquad\hbox{and}\cr
\sin(x+y) &= \sin x \cos y+\cos x\sin y. \cr} }$
\smallskip
$$\eqalign{
\cos(x+y)+i\sin(x+y) 
&= e^{i(x+y)} \cr
&= e^{ix+iy} = e^{ix}e^{iy} \cr
&= (\cos x+i\sin x)(\cos y+i\sin y) \cr
&= \cos x\cos y+ i\cos x\sin y + i\sin x \cos y + i^2\sin x\sin y \cr
&= \big(\cos x\cos y+(-1)\sin x\sin y\big)
+ i\big(\cos x\sin y+\sin x\cos y\big). \cr
}$$
Comparing terms on each side gives
$$\eqalign{
\cos(x+y)&=\cos x\cos y - \sin x\sin y, \qquad\hbox{and}\cr
\sin(x+y) &= \sin x \cos y+\cos x\sin y. \cr
}$$

\vfill\eject

\bigskip\bigskip
Define
$$\eqalign{
e^x &= 1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots, \cr
\sinh x &= x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}
+{x^9\over 9!}-{x^{11}\over 11!}+{x^{13}\over 13!} -\cdots, \cr
\cosh x &= 1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}
+{x^8\over 8!}-{x^{10}\over 10!}+{x^{12}\over 12!} -\cdots, \cr
}
$$
and
$$\tanh x ={\sinh x\over \cosh x}, \quad
\coth x ={1\over \tanh x}, \quad
\sech x ={1\over \cosh x}, \quad
\csch x ={1\over \sinh x}. 
$$


\bigskip\noindent
{\bf Example:}  Explain why $e^x = \cosh x+ \sinh x$. 
\bigskip
$$\eqalign{
e^x 
&= 1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots \cr
&= \left(1+{x^2\over 2!}+{x^4\over 4!} +{x^6\over 6!}+\cdots\right)
+\left(x+{x^3\over 3!}+{x^5\over 5!}+{x^7\over 7!} +\cdots\right) \cr
&= \cosh x + \sinh x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why $\cosh(-x)=\cosh x$ and $\sinh(-x)=-\sinh x$.
\bigskip
$$
\eqalign{
\cosh (-x) 
&= 1+{(-x)^2\over 2!}+{(-x)^4\over 4!}+{(-x)^6\over 6!}
+{(-x)^8\over 8!}+{(-x)^{10}\over 10!}+{(-x)^{12}\over 12!} +\cdots \cr
&= 1+{x^2\over 2!}+{x^4\over 4!}+{x^6\over 6!}
+{x^8\over 8!}+{x^{10}\over 10!}+{x^{12}\over 12!} +\cdots \cr
&=\cosh x, \cr
}$$
and
$$\eqalign{
\sinh(-x) 
&= (-x)+{(-x)^3\over 3!}+{(-x)^5\over 5!}+{(-x)^7\over 7!}
+{(-x)^9\over 9!}+{(-x)^{11}\over 11!}+{(-x)^{13}\over 13!} +\cdots \cr
&= -x-{x^3\over 3!}-{x^5\over 5!}-{x^7\over 7!}
-{x^9\over 9!}-{x^{11}\over 11!}-{x^{13}\over 13!} -\cdots \cr
&= -\left(x+{x^3\over 3!}+{x^5\over 5!}+{x^7\over 7!}
+{x^9\over 9!}+{x^{11}\over 11!}+{x^{13}\over 13!} +\cdots\right) \cr
&= -\sinh x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why 
$\displaystyle{\cosh x = {e^x+e^{-x}\over 2} }$
and $\displaystyle{ \sinh x = {e^x-e^{-x}\over 2} }$.
\bigskip
$$\eqalign{
\hbox{$1\over2$}(e^x + e^{-x}) 
&= \hbox{$1\over2$}\big(
1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots \cr
&\phantom{=\hbox{$1\over2$}}+
1+(-x)+{(-x)^2\over 2!}+{(-x)^3\over 3!}+{(-x)^4\over 4!}
+{(-x)^5\over 5!}+{(-x)^6\over 6!}+{(-x)^7\over 7!} +\cdots \big) \cr
&= \hbox{$1\over2$}\big(
1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots \cr
&\phantom{=\hbox{$1\over2$}}+
 1-x+{x^2\over 2!}-{x^3\over 3!}+{x^4\over 4!}
-{x^5\over 5!}+{x^6\over 6!}-{x^7\over 7!} +\cdots \big) \cr
&= 1+{x^2\over 2!}+{x^4\over 4!}+{x^6\over 6!}
+{x^8\over 8!}+\cdots \cr
&= \cosh x. \cr
}$$
$$\eqalign{
\hbox{$1\over2$}(e^x - e^{-x}) 
&= \hbox{$1\over2$}\big(
(1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots) \cr
&\phantom{=\hbox{$1\over2$}}-
(1+(-x)+{(-x)^2\over 2!}+{(-x)^3\over 3!}+{(-x)^4\over 4!}
+{(-x)^5\over 5!}+{(-x)^6\over 6!}+{(-x)^7\over 7!} +\cdots)\, \big) \cr
&= \hbox{$1\over2$}\big(
(1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots) \cr
&\phantom{=\hbox{$1\over2$}}-
(1-x+{x^2\over 2!}-{x^3\over 3!}+{x^4\over 4!}
-{x^5\over 5!}+{x^6\over 6!}-{x^7\over 7!} +\cdots )\, \big) \cr
&= \hbox{$1\over2$}\big(
1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots \cr
&\phantom{=\hbox{$1\over2$}}
-1+x-{x^2\over 2!}+{x^3\over 3!}-{x^4\over 4!}
+{x^5\over 5!}-{x^6\over 6!}+{x^7\over 7!} -\cdots \, \big) \cr
&= x+{x^3\over 3!}+{x^5\over 5!}+{x^7\over 7!}
+{x^9\over 9!}+\cdots \cr
&= \sinh x. \cr
}$$


\bigskip\noindent
{\bf Example:}  Explain why $\cosh^2 x-\sinh^2 x=1$.
\bigskip
$$\eqalign{
1 &= e^0 = e^{x+(-x)} \cr
&= e^x e^{-x} \cr
&= (\cosh x+\sinh x)(\cosh(-x)+\sinh(-x)) \cr
&= (\cosh x+\sinh x)(\cosh x-\sinh x)) \cr
&= \cosh^2 x - \sinh x\cosh x +  \sinh x\cosh x-\sinh^2 x \cr
&= \cosh^2 x-\sinh^2 x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Explain why\qquad
$\displaystyle{
\eqalign{
\cr
\cosh(x+y)&=\cosh x\cosh y + \sinh x\sinh y,
\qquad\hbox{and}\cr
\sinh(x+y) &= 2\sinh x \cosh y. \cr} }$
\smallskip
$$\eqalign{
\cosh x\cosh y + \sinh x\sinh y
&= \left({e^x+e^{-x}\over 2}\right)\left( {e^y+e^{-y}\over 2}\right)
+\left({e^x-e^{-x}\over 2}\right)\left( {e^y-e^{-y}\over 2}\right) \cr
&= {e^xe^y+e^{-x}e^y+e^xe^{-y}+e^{-x}e^{-y}\over 4}  \cr
&\qquad +{e^xe^y-e^{-x}e^y-e^xe^{-y}+e^{-x}e^{-y}\over 4} \cr
&= {2e^xe^y+2e^{-x}e^{-y}\over 4}  \cr
&= {e^{(x+y)}+e^{-(x+y)}\over 2}  \cr
&= \cosh (x+y).  \cr
}$$
and
$$ \eqalign{
\sinh x \cosh y+\cosh x\sinh y
&= \left({e^x-e^{-x}\over 2}\right)\left( {e^y+e^{-y}\over 2}\right) 
+\left({e^x+e^{-x}\over 2}\right)\left( {e^y-e^{-y}\over 2}\right) \cr
&= {e^xe^y-e^{-x}e^y+e^xe^{-y}-e^{-x}e^{-y}\over 4} \cr
&\qquad
+{e^xe^y+e^{-x}e^y-e^xe^{-y}-e^{-x}e^{-y}\over 4} \cr
&= {2e^{x+y}-2e^{-(x+y)}\over 4} \cr
&= \sinh (x+y). \cr}$$

\vfill\eject
\end