Question

Let the function $f(x)=\frac{arc\cot \frac{x}{2}+arc\cot \frac{x}{3}}{arc\tan \frac{x}{2}+arc\tan \frac{x}{3}}$, then $f(1)$ is equal to

Answer 1

Answer: 3

We have, $f(1)=\frac{{\cot }^{-1}\frac{1}{2}+{\cot }^{-1}\frac{1}{3}}{{\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3}}$
$=\frac{{\tan }^{-1}2+{\tan }^{-1}3}{{\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3}}$
$=\frac{\frac{3\pi }{\frac{4}{4}}}{4}=3$