Given: sin^-1(tanh(x)) = tan^-1(sinh(x))
Use the property u = sin^-1(sin(u)) on the right side and mark as equation [1]:
sin^-1(tanh(x)) = sin^-1(sin(tan^-1(sinh(x)))" [1]"
Digress and prove that sin(tan^-1(sinh(x)) = tanh(x)
An alternate form for tan^-1(u) = i/2ln(1-iu)-i/2ln(1+iu)
Substitute sinh(x) for u:
tan^-1(sinh(x)) = i/2ln(1-isinh(x))-i/2ln(1+isinh(x))
An alternate form for sin(v) = i/2e^(-iv)-i/2e^(iv)
sin(tan^-1(sinh(x))) = i/2e^(-i(i/2ln(1-isinh(x))-i/2ln(1+isinh(x)))-i/2e^(i(i/2ln(1-isinh(x))-i/2ln(1+isinh(x))))
Distribute through -i:
sin(tan^-1(sinh(x))) = i/2e^((-i^2/2ln(1-isinh(x))+i^2/2ln(1+isinh(x))))-i/2e^(i(i/2ln(1-isinh(x))-i/2ln(1+isinh(x))))
Distribute through i:
sin(tan^-1(sinh(x))) = i/2e^((-i^2/2ln(1-isinh(x))+i^2/2ln(1+i(sinh(x))))-i/2e^((i^2/2ln(1-isinh(x))-i^2/2ln(1+isinh(x))))
use the property i^2 = -1:
sin(tan^-1(sinh(x))) = i/2e^((1/2ln(1-isinh(x))-1/2ln(1+isinh(x))))-i/2e^((-1/2ln(1-isinh(x))+1/2ln(1+isinh(x))))
Write the 1/2 in the exponent as a square root:
sin(tan^-1(sinh(x))) = i/2e^((ln(sqrt(1-isinh(x)))-ln(sqrt(1+isinh(x)))))-i/2e^((-ln(sqrt(1-isinh(x)))+ln(sqrt(1+isinh(x)))))
Factor out i/2:
sin(tan^-1(sinh(x))) = i/2{e^((ln(sqrt(1-isinh(x)))-ln(sqrt(1+isinh(x))))-e^((-ln(sqrt(1-isinh(x)))+ln(sqrt(1+isinh(x)))))}
Use the property of logarithms ln(a) - ln(b) = ln(a/b):
sin(tan^-1(sinh(x))) = i/2{e^((ln((sqrt(1-isinh(x)))/(sqrt(1+isinh(x))))))-e^((ln((sqrt(1+isinh(x)))/(sqrt(1-isinh(x))))))}
Use the property e^ln(u) = u:
sin(tan^-1(sinh(x))) = i/2{(sqrt(1-isinh(x)))/(sqrt(1+isinh(x)))-(sqrt(1+isinh(x)))/(sqrt(1-isinh(x)))}
When we make a common denominator we obtain:
sin(tan^-1(sinh(x))) = i/2{(1-isinh(x))/(sqrt(1+sinh^2(x)))-(1+isinh(x))/(sqrt(1+sinh^2(x)))}
Combine over the common denominator:
sin(tan^-1(sinh(x))) = i/2{(-2isinh(x))/(sqrt(1+sinh^2(x)))}
The leading coefficient multiplied into the numerator becomes 1:
sin(tan^-1(sinh(x))) = sinh(x)/(sqrt(1+sinh^2(x)))
Use the identity 1 + sinh^2(x) = cosh^2(x):
sin(tan^-1(sinh(x))) = sinh(x)/(sqrt(cosh^2(x)))
sin(tan^-1(sinh(x))) = sinh(x)/cosh(x)
sin(tan^-1(sinh(x))) = tanh(x)
Substitute this into equation [1]:
sin^-1(tanh(x)) = sin^-1(tanh(x)) Q.E.D.
