Answer
The required value of $\cos 22.5{}^\circ $ is $\frac{\sqrt{2+\sqrt{2}}}{2}$.
Work Step by Step
We have to find the value of $\cos 22.5{}^\circ $; the formula is used which represents the cos identity. The term $\cos 22.5{}^\circ $ comes under the first quadrant where the value of all trigonometric functions is positive. And the value of the first quadrant is between $0{}^\circ \,\text{to}\,\text{90}{}^\circ $.
$\begin{align}
& \cos 22.5{}^\circ =\cos \frac{45{}^\circ }{2} \\
& =\sqrt{\frac{1+\cos 45{}^\circ }{2}} \\
& =\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}} \\
& =\sqrt{\frac{2+\sqrt{2}}{2}}
\end{align}$
Thus, the value of $\cos 22.5{}^\circ $ is $\frac{\sqrt{2+\sqrt{2}}}{2}$.
