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<b>let </b><font color="Maroon">th1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">th2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<span class="p1">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span> = 
<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_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2

<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_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</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"><span class="p5">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2

<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_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E2:33"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">th2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">th2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default"><font color="Maroon">th1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">th2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


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