$\frac{1-\tan ^{2} 22 \frac{1^{\circ}}{2}}{1+\tan ^{2} 22 \frac{1^{\circ}}{2}}$ is equal to :-

Step 1 ：Use the identity \(\tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x}\) to simplify the expression

Step 2 ：Calculate \(\cos 2x\) for \(x = 22.5^\circ\)

Step 3 ：Substitute the value of \(\cos 2x\) into the expression

Step 4 ：Simplify the expression to get the final answer

Step 5 ：\(\boxed{0.7071}\)