4 \(\cos \left ( \frac {x}{2}\right ) \)

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

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

Hence

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

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

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

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