Lagrange/Rolle question

Question

Prove that:

• for every $x>0:\arctan x+\arctan(1/x)=\pi/2$

• for $-1\leq x\leq1: \arcsin x+\arccos x=\pi/2$

This are some question I got on my Lagrange worksheet. Can someone help me with a solution with Lagrange/Rolle sentences?

Answer 1

Consider the function $$ f(x) = \arctan(x) + \arctan(1/x). $$ Its derivative is $$ f'(x)= \frac{1}{1+x^2} + \frac{1}{1+1/x^2} \frac{-1}{x^2} = \frac{1}{1+x^2}-\frac{1}{1+x^2}=0. $$ Therefore the function is constant over $(0,\infty)$. Then $f(x) = \lim_{x\to\infty} f(x) = \frac{\pi}{2}$.