Prove $\tan^{-1}(2)+\cot^{-1}(3)+\frac{1}{7} > \frac{\pi}{2}$

Question

I managed to reduce it to the form $$\cot(\frac{1}{7})<7$$ But I still don't know how to prove it

Answer 1

Note that

$$x -\tan^{-1}x=\int_0^x \frac{t^2}{1+t^2}dt>0$$

for $x> 0$. Then, let $x=\frac17$ to get $\frac17> \tan^{-1}\frac17=\cot^{-1}7$, or

$$\cot \frac17< 7$$