Another Hidden Trigonometric Identity

Pair up the products such that their arguments sum to $\frac{\pi}{4}$ , i.e., $(1+\tan\frac{\pi}{180})(1+\tan\frac{44\pi}{180})$ , $(1+\tan\frac{2\pi}{180})(1+\tan\frac{43\pi}{180})$ , etc. Let $x$ and $y$ be two such arguments. Expanding the product gives $(1+\tan x)(1+\tan y)=1+\tan x+\tan y+\tan x\tan y.$ From $\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$ , $1+\tan x+\tan y+\tan x\tan y=1+\tan(x+y)(1-\tan x\tan y)+\tan x\tan y.$ Because $x+y=\frac{\pi}{4}$ , $(1+\tan x)(1+\tan y)=1+1-\tan x\tan y+\tan x\tan y$ $=2.$ There are 22 such pairs, and $1+\tan\frac{45\pi}{180}=2$ , so the answer is $\boxed{2^{23}}$ , or $\boxed{8388608}$ .