5 \(\sin \left ( \frac {x}{2}\right ) \)

From the double angle formula (3C)

\[ \cos \left ( 2A\right ) =\cos ^{2}A-\sin ^{2}A \]

But \(\cos ^{2}A+\sin ^{2}A=1\) then \(\cos ^{2}A=1-\sin ^{2}A\) and the above becomes

\begin{align*} \cos \left ( 2A\right ) & =1-\sin ^{2}A-\sin ^{2}A\\ & =1-2\sin ^{2}A \end{align*}

Hence

\[ \sin ^{2}A=\frac {1-\cos \left ( 2A\right ) }{2}\]

Let \(A=\frac {x}{2}\) then the above becomes

\begin{align*} \sin ^{2}\left ( \frac {x}{2}\right ) & =\frac {1-\cos \left ( x\right ) }{2}\\ \sin \left ( \frac {x}{2}\right ) & =\pm \sqrt {\frac {1-\cos \left ( x\right ) }{2}}\end{align*}

The sign depends on the quadrant of \(\frac {x}{2}\).