ifsin-12a/1+a2- cos-11-b2/1+b2 = tan-12x/1-x2 then prove that x = a-b/1+ab - Maths - Inverse Trigonometric Functions

Question

ifsin-12a/1+a2-
cos-11-b2/1+b2 =
tan-12x/1-x2 then prove that x = a-b/1+ab

Varun.Rawat · Accepted Answer

sin-12a1+a2 - cos-11-b21+b2 = tan-12x1-x2Put a = tan θ; b = tan α  and  x = tan βNow, sin-12 tan θ1 + tan2θ - cos-11-tan2α1+tan2α = tan-12tan β1-tan2β⇒sin-1sin 2θ - cos-1cos 2α = tan-1tan 2β⇒2θ - 2α = 2β⇒θ - α = β⇒tan-1a - tan-1b = tan-1x⇒tan-1a - b1 + ab= tan-1x⇒a-b1+ab = x

ifsin-12a/1+a2- cos-11-b2/1+b2 = tan-12x/1-x2 then prove that x = a-b/1+ab.

ifsin^-12a/1+a²- cos^-11-b²/1+b²= tan^-12x/1-x²then prove that x = a-b/1+ab.