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      <text>5.2.18 (Half angle formulas)</text>
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      <text><![CDATA[<h3> Half-angle formulae </h3>

<p>If we replace \(x\) by \(\dfrac{x}{2}\) and, consequently \(2x\) by \(x\) in the double angle formula, we obtain 
\[
\sin x\equiv 2\sin\left(\dfrac{x}{2}\right)\cos\left(\dfrac{x}{2}\right) 
\]</p>
<br/>
<p>Similarly for the cosine:
\[
\cos x\equiv 2\cos^2\left(\dfrac{x}{2}\right)-1
\]</p>

<p>These are examples of <b>half-angle formulae</b>. </p>

<p>We can obtain a half-angle formula for \(\tan x\) as well. Replacing \(x\) by <span \(\dfrac{x}{2}\) and \(2x\) by \(x\) in the double angle formula for \(\tan\) we obtain:
\[
\tan x \equiv \dfrac{2\tan\left(\dfrac{x}{2}\right)}{1-\tan^2\left(\dfrac{x}{2}\right)}
\]
<h4>Key Point<br></h4>
<p>If <span class="math inline">\(t= \tan\left(\dfrac{A}{2}\right)\)</span> then <br>\(\sin a = \dfrac{2t}{1+t^2}\)
<br> \(\cos A= \dfrac{1-t^2}{1+t^2}\)
<br>\(\tan A= \dfrac{2t}{1-t^2}\)</p>]]></text>
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