求证arctan1+arctan2+arctan3=pai

题目

求证arctan1+arctan2+arctan3=pai

答案

证明:
设arctan1+arctan2+arctan3=x
那么tanx=tan(arctan1+arctan2+arctan3)
=(tan(arctan1+arctan2)+tan(arctan3))/(1-tan(arctan1+arctan2)tan(arctan3)
=(tan(arctan1+arctan2)+3)/(1-tan(arctan1+arctan2)*3)
又tan(arctan1+arctan2)=(tan(arctan1)+tan(arctan2))/(1-tan(arctan1)*tan(arctan2))=(1+2)/(1-2) =-3
所以tanx=(-3+3)/(1-(-3)*3)=0
而arctan1 arctan2 arctan3 都是锐角
故有x=π,即arctan1+arctan2+arctan3=π