<s>[INST] Why is $(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000\cdots$? Apparently, 
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.
Why is that?
 [/INST] Alternatively,
$$\cot(x+y) = \frac{\cos(x+y)}{\sin(x+y)} = \frac{\cos x \cos y - \sin x \sin y}{\sin x \cos y + \cos x \sin y} = \frac{\cot x \cot y - 1}{\cot x + \cot y}$$
Now if $x = 37^\circ$, $y = 8^\circ$, then $\cot(x + y) = \cot 45^\circ = 1$ and
$$\begin{align}  \ \ \ \  1 & = \frac{\cot x \cot y - 1}{\cot x + \cot y} \\  \cot x + \cot y & = \cot x \cot y - 1 \\
   2 + \cot x + \cot y & = \cot x \cot y + 1 \\
 1 - \cot x - \cot y + \cot x \cot y  & = 2 \\
(1 - \cot x)(1 - \cot y) & = 2 \\
(1 - \cot 37^\circ)(1 - \cot 8^\circ) & = 2
\end{align}$$
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