Question

Prove that sin `[tan^-1 ((1 - x^2)/(2x)) + cos^-1 ((1 - x^2)/(1 + x^2))]` = 1

Answer 1

L.H.S. = sin `[tan^-1 ((1 - x^2)/(2x)) + cos^-1 ((1 - x^2)/(1 + x^2))]`

Substituting x = tan θ, we get

L.H.S. = sin `[tan^-1 ((1 - tan^2theta)/(2tantheta)) + cos^-1 ((1 - tan^2theta)/(1 + tan^2theta))]`

= `sin[tan^-1 (1/tan 2theta) + cos^(-1) (cos 2theta)]`

= `sin[tan^-1 (cot 2theta) + cos^-1 (cos 2theta)]`

= `sin[tan^-1 {tan (pi/2 - 2theta)} + 2theta]`

= `sin(pi/2 - 2theta + 2theta)`

= `sin(pi/2)`

= 1

= R.H.S.