Question

need help finding the inverse of this function. f(x)=2ln(sin(x))-ln(1-cos(x))

Answer 1

Simplify\[\large f(x)=2\ln(\sin(x))-\ln(1-\cos(x))\]\[\large = \ln(\sin(x)^2)-\ln(1-\cos(x))\]\[\large = \ln(\frac{ \sin(x)^2 }{ 1 - \cos(x) } )\]\[\large = \ln(\frac{ 1 - \cos(x)^2 }{ 1 - \cos(x) })\]\[\large = \ln (\frac{ (1 + \cos (x))(1 - \cos (x)) }{ 1 - \cos(x) })\]\[\large= \ln(1 + \cos x)\]
Find x\[\large f(x) = \ln(1 + \cos(x)) \]\[\large e^{f(x)} = 1 + \cos (x)\]\[\large e^{f(x)} - 1 = \cos (x)\]\[\large \cos^{-1}(e^{f(x)} - 1) = x\]
Rename/answer\[f^{-1}(x) = \large \cos^{-1}(e^{x} - 1)\]