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      <text>4.3.3.7 Sum of sines and cosines</text>
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      <text><![CDATA[<h4> Sum of two sines and sum of two cosines </h4>

<p>Here we obtain results that are widely used in areas of science and engineering such as vibration theory, wave theory and electric circuit theory.</p>
<p>We return to the identities angle addition formulae:</p>
\[\begin{aligned}
\sin(A+B ) &amp;\equiv \sin A\cos B+\cos A\sin B\\
\sin (A-B) &amp;\equiv \sin A\cos B-\cos A\sin B
\end{aligned}\]

<p>Adding these identities gives</p>
\[\sin(A+B)+\sin(A-B)\equiv 2\sin A\cos B.\]
<p>Subtracting the identities produces</p>
\[\sin (A+B)-\sin(A-B)\equiv 2\cos A\sin B\]

<p>It is now convenient to let <span class="math inline">\(C=A+B\) </span>and<span class="math inline"> \(D=A-B\)</span> so that</p>
\[A=\dfrac{C+D}{2} \qquad \text{and} \qquad B=\dfrac{C-D}{2}\]
<p>The above then becomes:</p>
\[\sin C+\sin D\equiv 2\sin\left(\dfrac{C+D}{2}\right)\cos \left(\dfrac{C-D}{2}\right) \]
<p>Similarly:</p>
\[\sin C-\sin D\equiv 2\cos\left(\dfrac{C+D}{2}\right)\sin \left(\dfrac{C-D}{2}\right) \]]]></text>
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