Two variable trigonometric inequality

Question

Let $x>3$ and $k>0$ be both reals. I want to show algebraically that If $k\in(0,1)$, then $k\cot(x)>\cot(\frac{x}{k})$. If $k>1$, then $k\cot(x)<\cot(\frac{x}{k})$.It is clear that the expression is $0$ if $k=1$. Graphically and with Taylor expansion at infinity I can see it, but I was trying to prove this algebraically. Thanks for your time.