Question

Find the value of the expression sin(2 tan -1 1/3) + cos(tan -1 2 [root 2]).

Answer 1

$let\ Tan^{-1}\frac{1}{3}=x,\ \ \ tanx=1/3,\ \ Sin2x=\frac{2/3}{1-1/3^2}=3/4\\\\Tan^{-1}2\sqrt2=y,\ \ tany=2\sqrt2,\ \ sec^2y=9,\ Cosy=1/3\\\\Sin(2\ tan^{-1}\frac{1}{3})+Cos(Tan^{-1} \sqrt2)\\\\=3/4\ + \ 1/3=13/12$