<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<i><font color="Green">E12</font></i>:  <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3">cosech</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos5.html#D3">SIN_COS5:def 3</a></i>
;<br/>
<i><font color="Green">E13</font></i>:  <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K2">sech</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos5.html#D2">SIN_COS5:def 2</a></i>
;<br/>
 <a href="sin_cos2.html#K9">tanh</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos8.html#T1" target="_self">Th1</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos5.html#D1">SIN_COS5:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:10"/>
(  <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3">cosech</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K2">sech</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="sin_cos2.html#K9">tanh</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos8.html#E1:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="sin_cos8.html#E2:10"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
