Prove that Arcsin(Tanh(x)) = Arctan(Sinh(x))?

Answer 1

Let say
y = arcsin(tanh(x))
sin(y) = tanh(x)
sin(y) = sinh(x) / cosh(x)
sin(y) = [e^x - e^(-x)] / [e^x + e^(-x)]

cos^2(y) = 1 - sin^2(y)
cos^2(y) = 1 - [e^x - e^(-x)]^2 / [e^x + e^(-x)]^2
cos^2(y) = ([e^x + e^(-x)]^2 - [e^x - e^(-x)]^2) / [e^x + e^(-x)]^2
cos^2(y) = 4 / [e^x + e^(-x)]^2
cos(y) = 2 / [e^x + e^(-x)]^2

tan(y) = sin(y)/cos(y)
tan(y) = [e^x - e^(-x)]/[e^x + e^(-x)] divided by 2/[e^x + e^(-x)]^2
tan(y) = [e^x - e^(-x)] / 2
tan(y) = sinh(x)
y = arctan(sinh(x))

thus
Arcsin(Tanh(x)) = Arctan(Sinh(x))
Proved!!!

Answer 2

line 9 is wrong ..