TITLE: A "fast" way for  computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?
QUESTION [19 upvotes]: Which  is the fastest paper-pencil approach to compute the product $$ \prod
 \limits_{i=1}^{45}(1+\tan i^\circ) $$

REPLY [4 votes]: Using this, $$(\cot A + \tan y)(\cot A+ \tan(A-y))=\csc^2A \text{ if } A\ne m\pi\text{ where }m\text{ is any integer}$$
Putting $A=45^\circ, (1 + \tan y)(1+ \tan(45^\circ-y))=\csc^245^\circ=2$
Now, putting $y=1^\circ,2^\circ,3^\circ,\cdots,\lfloor\frac{45}2\rfloor^\circ=22^\circ$ and multiplying them we get, $$\prod_{1\le r\le 22}(1+\tan r^\circ)(1+\tan(45-r)^\circ)=2^{22}$$
$$\implies   \prod_{1\le r\le 44}(1+\tan r^\circ)=2^{22}$$
The unpaired $1+\tan45^\circ=1+1=2$