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

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E20</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

 <a href="sin_cos4.html#K2">cot</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="sin_cos4.html#K2">cot</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos4.html#T18">SIN_COS4:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:19"/>
 <a href="sin_cos4.html#K2">cot</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K3">|^</a> 3</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos4.html#K2">cot</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 ;<br/>


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