Is there a real number $x$, that the expressions $\tan x + \sqrt{3}$ and $\cot x+ \sqrt{3}$ are both integers?

My solution:

First, let's assume $\tan x + \sqrt{3}=a$ and $\cot x+ \sqrt{3}=b$ where $a,\,b$ are both integers.

From the second equality, we get:

$\cot x+ \sqrt{3}=b$

$\cot x=b-\sqrt{3}$

$\dfrac{1}{\tan x}=b-\sqrt{3}$

$\tan x=\dfrac{1}{b-\sqrt{3}}$

so we get

$\tan x=\dfrac{1}{b-\sqrt{3}}=a-\sqrt{3}$

$(a-\sqrt{3})(b-\sqrt{3})=1$

$\sqrt{3}=\dfrac{ab+2}{a+b}$

But recall that both $a,\,b$ are both integers, this last equality contradicts to our previous assumption and so the answer is nope.

There isn't any real x that serves both the given two expressions to be integers.

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