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.