Answer
$-\displaystyle \frac{\sqrt{42}}{12}$
Work Step by Step
Letting $A=2x, \displaystyle \quad\cos A=-\frac{5}{12}$
we express $x=\displaystyle \frac{A}{2}, \displaystyle \quad\cos x=\cos\frac{A}{2}$
($\cos x$ is negative as x terminates in Q.II.)
Half-Angle identity:
$\displaystyle \cos x=\cos\frac{A}{2}=-\sqrt{\frac{1+\cos A}{2}}$
$=-\sqrt{\dfrac{1+(-\dfrac{5}{12})}{2}}$
$=-\sqrt{\dfrac{12-5}{24}}=-\sqrt{\dfrac{7}{24}}$
$=-\displaystyle \dfrac{\sqrt{7}}{2\sqrt{6}}\cdot\dfrac{\sqrt{6}}{\sqrt{6}}=-\dfrac{\sqrt{42}}{12}$
