Answer
The value is $\frac{\sqrt{3}-1}{2\sqrt{2}}$.
Work Step by Step
Use the equation:
$\cos 75{}^\circ =\cos \left( 120{}^\circ -45{}^\circ \right)$
Applying the formula $\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b $
We get:
$\begin{align}
& \cos \left( 120{}^\circ -45{}^\circ \right)=\cos 120{}^\circ \cos 45{}^\circ +\sin 120{}^\circ \sin 45{}^\circ \\
& =-\frac{1}{2}\times \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{2}} \\
& =\frac{\sqrt{3}-1}{2\sqrt{2}}
\end{align}$
Hence, $\cos 75{}^\circ =\frac{\sqrt{3}-1}{2\sqrt{2}}$.
