sin[ tan^-1 x (1-x^2/2x) + + cos^-1 (1-x^2/1+x^2)]=1pls prove this - Maths - Inverse Trigonometric Functions

Question

sin[ tan^-1 x (1-x^2/2x) + + cos^-1 (1-x^2/1+x^2)]=1pls prove
this

Vipin Verma · Accepted Answer

sin[tan-11-x22x+ cos-11-x21+x2] Let x = tanθ So tan-11-x22x =
tan-11-tan2θ2tanθ =tan-1
[cos2θ-sin2θ2sinθcosθ] =tan-1[cot2θ] =
tan-1[tan(π2-2θ)] =π2-2θcos-11-x21+x2
=cos-11-tan2θ1+tan2θ
=cos-1[cos2θ-sin2θcos2θ+sin2θ]
=cos-1[cos2θ] = 2θsin[tan-11-x22x+ cos-11-x21+x2]
=sin[π2-2θ +2θ] =sinπ2 = 1