Question

Prove that 2tan^-1(1/2)+tan^-1(1/7)=sin^-1(31/25√2)

Answer 1

Answer:

As 2 tan^-1 x=tan^-1(2 x/1-x²)

tan^-1(2×1/2 / 1-(1/2)²) +tan^-1(1/7) =sin^-1(31/25√2)

tan^-1(4/3) +tan^-1(1/7) =sin^-1(31/25√2)

tan^-1{(4/3+1/7)/(1-4/3×1/7)} =sin^-1(31/25√2)

tan^-1(31/17) =sin^-1(31/25√2)

RHS

By pyth. theorem

(25√2)²=31²+x²

x=17

sin^-1(31/25√2)= tan^-1(31/17)

∴LHS =RHS

HP