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    <name>
      <text>6.2.2.5 (not in quiz) sinh(x+y)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise&nbsp;</p>
<p>In this exercise, we show that \(\sinh(x+y) \equiv \sinh x \cosh y + \cosh x \sinh y\).</p>
<ol class="HELM_parts">
<li><p> First, express \(\sinh x \cosh y \) in terms of exponentials.</p><p>\(\sinh x \cosh y&nbsp; \equiv \) [[input:ans1]] [[validation:ans1]]</p></li>
<li><p> Next, express \(\cosh x \sinh y \) in terms of exponentials.</p><p>\(\cosh x \sinh y&nbsp; \equiv \) [[input:ans2]] [[validation:ans2]]</p></li>
<li><p> Now, express \(\sinh x \cosh y + \cosh x \sinh y\) in terms of exponentials using (a) and (b),</p><p>\(\sinh x \cosh y + \cosh x \sinh y \equiv \) [[input:ans3]] [[validation:ans3]]</p></li>
<p>and realize that this equals&nbsp;\(\sinh(x+y)\) by definition.</p>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>(a) By definition of hyperbolic functions, we have \( \sinh x \cosh y \equiv \left( {@sinhx@} \right) \cdot \left( {@coshy@} \right) \equiv {@m1@} \).</p>
<p>(b) Similarly, we have \( \cosh x \sinh y \equiv \left( {@coshx@} \right) \cdot \left( {@sinhy@} \right) \equiv {@m2@} \).</p>
<p>(c) From (a) and (b), we deduce \[ \begin{align*} &amp; \sinh x \cosh y + \cosh x \sinh y \\ &amp; = {@m1@} + {@m2@} \\ &amp; = {@m3@} \\ &amp; = \sinh (x+y). \end{align*}\]</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2020070100</text>
    </stackversion>
    <questionvariables>
      <text>coshx:1/2*(e^x + e^(-x));
coshy:1/2*(e^y + e^(-y));
sinhx:1/2*(e^x - e^(-x));
sinhy:1/2*(e^y - e^(-y));
simp:true;
m1: expand(sinhx*coshy);
m1: combine(m1);
m2: expand(coshx*sinhy);
m2: combine(m2);
m3: combine(m1+m2);
m3: expand(radcan(m3));
m3: combine(m3);</text>
    </questionvariables>
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      <text><![CDATA[[[feedback:prt1]]<p>[[feedback:prt2]]<br></p><p>[[feedback:prt3]]<br></p>]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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