Question

If `sin^-1  (2a)/(1+a^2)+sin^-1  (2b)/(1+b^2)=2tan^-1x,` Prove that  `x=(a+b)/(1-ab).`

Answer 1

Let: a = tan z
      b = tan y

Then,

`sin^-1  (2a)/(1+a^2)+sin^-1  (2b)/(1+b^2)=2tan^-1x`

`=>sin^-1  (2tanz)/(1+tan^2z)+sin^-1  (2tany)/(1+tan^2y)=2tan^-1x`

`=>sin^-1(sin2z)+sin^-1(sin2y)=2tan^-1x`       `[becausesin2x=(2tanx)/(1+tan^2x)]`

`=>2z+2y=2tan^-1x`

`=>tan^-1a+tan^-1b=tan^-1x`       `[becausea=tanzandb=tany]`

`=>tan^-1  (a+b)/(1-ab)=tan^-1x`     `[becausetan^-1x+tan^-1y=tan^-1  (x+y)/(1-xy)]`

`=>x=(a+b)/(1-ab)`