Question

If `sin^-1  (2a)/(1+a^2)-cos^-1  (1-b^2)/(1+b^2)=tan^-1  (2x)/(1-x^2)`, then prove that `x=(a-b)/(1+ab)`

Answer 1

Let: a = tan m 
      b = tan n 
      x = tan y

Now,

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

`=>sin^-1  (2tanm)/(1+tan^2m)-cos^-1  (1-tan^2n)/(1+tan^2n)=tan^-1  (2tany)/(1-tan^2y)`

`=>sin^-1(sin2m)-cos^-1(cos2n)=tan^-1(tan2y)`      `[becausesin2x=(2tanx)/(1+tan^2x)andcos2x=(1-tan^2x)/(1+tan^2x)]`

`=>2m-2n=2y`

`=>m-n=y`

`=>tan^-1a-tan^-1b=tan^-1x`      `[becausea=tanm,b=tannandx=tany]`

 

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

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