Product of tans

$\quad[\tan(x)+1][\tan(45^\circ-x)+1]$

$=[\tan(x)+1]\left[\dfrac{1-\tan x}{1+\tan x}+1\right]$

$=[\tan(x)+1]\left[\dfrac{2}{1+\tan x}\right]$

$=2$

There are 22 pairs of such angles, and including the case for $45^\circ$ , we would have $\log_22^{23}$ as the answer, which is $23$ .

We can have:

$1+\tan {r^{\circ}} = \frac{\cos r^{\circ} + \sin r^{\circ}}{\cos {r^{\circ}}} \\ = \frac{\sqrt2 \cos ( 45^{\circ} -r^{\circ})}{\cos r^{\circ}} ( \text{Trigonometric R Method})$

So,

\( \begin{array}{} \log_{2}{[\prod_{r=1}^{45} (1+\tan r^{\circ})]} \\ = \log_{2}[\prod_{r=1}^{45}\frac{\sqrt2 \cos ( 45^{\circ} -r^{\circ})}{\cos r^{\circ}} ]\\=\log_{2}[\frac{(\sqrt2) ^{45} \cos ( 44^{\circ}) \cos ( 43^{\circ}) \cos ( 42^{\circ})\cdots \cos ( 1^{\circ} )\cos ( 0^{\circ} )}{\cos ( 1^{\circ}) \cos ( 2^{\circ}) \cos ( 3^{\circ})\cdots \cos ( 44^{\circ} )\cos ( 45^{\circ} )}] \\=\log_{2}[\frac{(\sqrt2) ^{45} \cos ( 1^{\circ}) \cos ( 2^{\circ}) \cos ( 3^{\circ})\cdots \cos ( 44^{\circ} )\cos ( 0^{\circ} )}{{ }\cos ( 1^{\circ}) \cos ( 2^{\circ}) \cos ( 3^{\circ})\cdots \cos ( 44^{\circ} )\cos ( 45^{\circ} )}]\\=\log_{2}[\frac{(\sqrt2) ^{45} }{\frac{\sqrt 2}{2}}]\\ =\boxed{23}\end{array} \)