Answer
The exact value of $\cos \left( 120{}^\circ -45{}^\circ \right)$ is $\frac{-\sqrt{2}+\sqrt{6}}{{4}}$.
Work Step by Step
Use the difference formula of cosines and evaluate the term as,
$\cos \left( 120{}^\circ -45{}^\circ \right)=\cos 120{}^\circ \cos 45{}^\circ +\sin 120{}^\circ \sin 45{}^\circ $
Substitute the values $\cos 120{}^\circ =-\frac{1}{2},\text{ }\cos 45{}^\circ =\frac{\sqrt{2}}{{2}},\text{ }\sin 120{}^\circ =\frac{\sqrt{3}}{2},\text{ and }\sin 45{}^\circ =\frac{\sqrt{2}}{{2}}$.
$\begin{align}
& \cos \left( 120{}^\circ -45{}^\circ \right)=\left( -\frac{1}{2}\times \frac{\sqrt{2}}{{2}} \right)+\left( \frac{\sqrt{3}}{2}\times \frac{\sqrt{2}}{{2}} \right) \\
& =\frac{-\sqrt{2}+\sqrt{6}}{4}
\end{align}$
Hence, the exact value of $\cos \left( 120{}^\circ -45{}^\circ \right)$ is equivalent to $\frac{-\sqrt{2}+\sqrt{6}}{4}$.
