If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not? It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be removed by doing that! Can you help me with proving that $a=b$ or not?
Actually I'm going to prove that $x+\sqrt{x^2+1}$ is a $1$-$1$ function.

Hint:
WLOG $a=\cot2A,b=\cot2B,0<A,B<\dfrac\pi2$
we have $\cot  A=\cot B\implies\cot2A=?$
