What is the value of cos 22 1 2 ° =

Let $\cos A=\cos {\left(22\frac{1}{2}\right)}^{\circ }$

$\Rightarrow A=22{\frac{1}{2}}^{\circ }$

Using formula, $\cos 2A=2{\cos }^{2}A-1$

$\Rightarrow \cos {45}^{\circ }=2{\cos }^{2}A-1$

$\Rightarrow \frac{1}{\sqrt{2}}+1=2{\cos }^{2}A$

$\Rightarrow 2{\cos }^{2}A=\left(\frac{1+\sqrt{2}}{\sqrt{2}}\right)$

$\Rightarrow \cos A=\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}$

Hence, $\cos A=\cos {\left(22\frac{1}{2}\right)}^{\circ }=\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}$$$