Question

4 arctan(1/3)+4 arctan(1/4)+4 arctan(2/9)=pi
how do i prove it but without using this formula
arctan(x)+arctan(y) = arctan( x+y/1+xy)

Answer 1

let A = arctan(1/3) , let B = arctan(1/4), let C = arctan(2/9)
tanA = 1/3 , tanB = 1/4, and tanC = 2/9
tan(A+B) = (tanA + tanB)/(1 -tanAtanB)
= (1/3 + 1/4)/(1 - (1/3)(1/4))
= (7/12)/(11/12) = 7/11

tan(A+B + C) = (tan(A+B) + tanC )/(1 - tan(A+B)tanC)
= (7/11 + 2/9)/(1 - (7/11)(2/9))
= (85/99) / (85/99)
= 1

IF tan(A+B+C) = 1
then A+B+C = π/4 , yeahhh, since ....

LS
= 4 arctan(1/3)+4 arctan(1/4)+4 arctan(2/9)
= 4(arctan(1/3) + arctan(1/4) + arctan(2/9)
= 4(A+B+C)
= 4(π/4) = π = RS

btw, the formula that I am not supposed to use, and I didn't , should have been
arctan(x)+arctan(y) = arctan( (x+y)/(1-xy) )