Prove that `tanpi/(16)=sqrt(4+2sqrt(2))-(sqrt(2)+1)`

Explanation

To prove that \( \tan\left(\frac{\pi}{16}\right) = \sqrt{4 + 2\sqrt{2}} - (\sqrt{2} + 1) \), we will follow these steps:

### Step 1: Find \( \tan(22.5^\circ) \)
We know that:
\[ \tan(22.5^\circ) = \tan\left(\frac{\pi}{8}\right) \]
Using the half-angle formula:
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