Answer
The exact value of $\cos \left( 45{}^\circ -30{}^\circ \right)$ is $\frac{1+\sqrt{3}}{2\sqrt{2}}$.
Work Step by Step
Use the difference formula of cosines and evaluate the term as,
$\cos \left( 45{}^\circ -30{}^\circ \right)=\cos 45{}^\circ \cos 30{}^\circ +\sin 45{}^\circ \sin 30{}^\circ $
Substitute the values $\cos 45{}^\circ =\frac{1}{\sqrt{2}},\text{ }\cos 30{}^\circ =\frac{\sqrt{3}}{2},\text{ }\sin 45{}^\circ =\frac{1}{\sqrt{2}},\text{ and }\sin 30{}^\circ =\frac{1}{2}$.
$\begin{align}
& \cos \left( 45{}^\circ -30{}^\circ \right)=\left( \frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2} \right)+\left( \frac{1}{\sqrt{2}}\times \frac{1}{2} \right) \\
& =\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}} \\
& =\frac{1+\sqrt{3}}{2\sqrt{2}}
\end{align}$
Hence, the exact value of $\cos \left( 45{}^\circ -30{}^\circ \right)$ is equivalent to $\frac{1+\sqrt{3}}{2\sqrt{2}}$.
