how to i calculate 22.5 degrees using tan?

You could also use the double angle formula:

$$\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}$$

If θ = 22.5° then 2θ = 45° and tan(45°) = 1 so you have:  

$$1=\frac{2\tan(22.5)}{1-\tan^2(22.5)}$$

This is a quadratic in tan(22.5°) the positive solution of which is √2 - 1 or 0.4142....

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Your question makes no sense however

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{22.5}}^\circ\right)} = {\mathtt{0.414\: \!213\: \!562\: \!373}}$$