Question

Prove that

`sin{tan^-1  (1-x^2)/(2x)+cos^-1  (1-x^2)/(2x)}=1`

Answer 1

`sin{tan^-1  ((1-x^2)/(2x))+cos^-1 ((1-x^2)/(1+x^2))}=1`

LHS = `sin{tan^-1  ((1-x^2)/(2x))+cos^-1  ((1-x^2)/(1+x^2))}`

`=sin{sin^-1(((1-x^2)/(2x))/sqrt(1+(1-x^2)/(2x)))+cos^-1((1-x^2)/(1+x^2))}`    `[becausetan^-1x=sin^-1  x/sqrt(1+x^2)]`

`=sin{sin^-1((1-x^2)/(1+x))+cos^1((1-x^2)/(1+x^2))}`

`=sin{pi/2}`      `[becausesin^-1x+cos^-1x=pi/2]`

 = 1 = RHS