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

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

<b>thus </b><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35" target="_self">PI</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font>

;<br/>

<b>thus </b><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35" target="_self">PI</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font>

;<br/>

<b>thus </b><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th</font></span>)</span> = 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th</font></span>)</span> = 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
0 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font>

;<br/>

<b>thus </b><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font>

;<br/>

<b>thus </b><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <a href="sin_cos.html#K35" target="_self">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <a href="sin_cos.html#K35" target="_self">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>

;<br/>


</div>
