求证arctan1+arctan2+arctan3=pai

1个回答

  • 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