Answer
See the full explanation below.
Work Step by Step
Let us consider the left side of the given expression,
$\cos \,2\alpha $
By using the identity of trigonometry,
$\cos \,\left( \alpha +\beta \right)=\cos \,\alpha \,\cos \beta -\sin \,\alpha \,\sin \,\beta $
Now, the above expression can be further simplified as,
$\begin{align}
& \cos \,2\alpha =\cos \,\left( \alpha +\alpha \right) \\
& =\cos \,\alpha \,\cos \,\alpha -\sin \,\alpha \,\sin \,\alpha \\
& ={{\cos }^{2}}\,\alpha -{{\sin }^{2}}\,\alpha
\end{align}$
Thus, the left side of the given expression is equal to the right side, which is
$\cos \,2\alpha ={{\cos }^{2}}\,\alpha -{{\sin }^{2}}\,\alpha $.
