Misc 9 - Prove tan-1 root x  = 1/2 cos-1 (1 - x)/(1 + x) - Not clear how to approach

  1. Chapter 2 Class 12 Inverse Trigonometric Functions
  2. Serial order wise
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Misc 9 Prove tan-1 x = 1/2 cos-1 ((1 x)/(1 + x)), x [0, 1] Taking R.H.S. 1/2 cos-1 ((1 x)/(1 + x)) Putting x = tan2 = 1/2 cos-1 ((1 tan2 )/(1 + tan2 )) = 1/2 cos-1 ((1 (sin2 )/(cos2 ))/(1 + (sin2 )/(cos2 ))) = 1/2 cos-1 (((cos2 sin2 )/(cos2 ))/((cos2 + sin2 )/(cos2 ))) = 1/2 cos-1 ((cos2 sin2 )/(cos2 ) (cos2 )/(cos2 + sin2 )) = 1/2 cos-1 ((cos2 sin2 )/(cos2 + sin2 ) ) = 1/2 cos-1 ((cos2 sin2 )/1 ) = 1/2 cos-1 (cos 2 ) = 1/2 2 = We assumed that x = tan2 = tan tan -1 = Hence, 1/2 cos-1 ((1 x)/(1 + x)) = 1/2 cos-1 ((1 x)/(1 + x)) = tan-1 x. Hence, R.H.S. = L.H.S. Hence Proved

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