Mathematics

Problem

Let

x>0

. a. (i) Prove that

\arctan x+\arctan\left(\dfrac{1}{x}\right)=\dfrac{\pi}{2}

. b. (ii) Using the result of part (i), evaluate

S=\arctan 2+\arctan\dfrac{1}{2}+\arctan 3+\arctan\dfrac{1}{3}.