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      <text>4.3.1.5 Sum of angles formula</text>
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      <text><![CDATA[<h4>Sum of angles identity</h4>
<p>This Key Point introduces two further important identities.</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
\[\begin{aligned}
\sin (A+B) &amp;\equiv \sin A\cos B+\cos A\sin B \qquad \qquad (1) \\
\cos(A+B) &amp;\equiv \cos A\cos B-\sin A\sin B \qquad \qquad (2)
\end{aligned}\]
</div>

<p>Note carefully the addition sign in \((1)\) but the subtraction sign in \((2)\). Further identities can readily be obtained from \((1)\) and \((2)\).</p>

<p>Dividing \((1)\) by \((2)\) we obtain</p>
\[\tan(A+B)\equiv \dfrac{\sin(A+B)}{\cos(A+B)}\equiv \dfrac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}\]
<p>Dividing every term by \(\cos A\ \cos B\) we obtain:</p>
\[\tan (A+B)\equiv \dfrac{\tan A+\tan B}{1-\tan A\tan B}\]
<p>Replacing \(B\) by \(-B\) in \((1)\) and \((2)\) and remembering that \(\cos (-B)\equiv \cos B\) and \(\sin(-B)\equiv -\sin B\) we find</p>
\[\begin{aligned}
\sin(A-B) &amp;\equiv \sin A\cos B-\cos A\sin B \\
\cos (A-B) &amp;\equiv \cos A\cos B+\sin A\sin B.
\end{aligned}\]]]></text>
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