Question

tan - _ + tan-I + tan- _ + tan

Answer 1

We have to prove that tan −1 1 5 + tan −1 1 7 + tan −1 1 3 + tan −1 1 8 = π 4 .

Solve left hand side.

tan −1 1 5 + tan −1 1 7 + tan −1 1 3 + tan −1 1 8 = tan −1 ( 1 5 + 1 7 1− 1 5 × 1 7 )+ tan −1 ( 1 3 + 1 8 1− 1 3 × 1 8 ) = tan −1 ( 7+5 35−1 )+ tan −1 ( 8+3 24−1 ) = tan −1 ( 12 34 )+ tan −1 ( 11 23 ) = tan −1 ( 6 17 )+ tan −1 ( 11 23 )

Simplify further,

tan −1 1 5 + tan −1 1 7 + tan −1 1 3 + tan −1 1 8 = tan −1 ( 6 17 + 11 23 1− 6 17 × 11 23 ) = tan −1 325 325 = tan −1 1 = π 4

It is equal to the right hand side.

Hence, it is proved that tan −1 1 5 + tan −1 1 7 + tan −1 1 3 + tan −1 1 8 = π 4 .