Algebra Level 4

Which of the following statements is/are correct?

$$[1]$$ An irrational number times a rational number is always an irrational number.

$$[2]$$ The statement $$\tan^{-1} x + \tan^{-1} y=\tan^{-1}\left( \frac{x+y}{1-xy}\right)$$ is true for all real numbers $$x$$ and $$y$$.

$$[3]$$ If $$x^y=y^x$$, then $$x=y$$ where $$x$$ and $$y$$ are positive real numbers.

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