Question

How do you prove arcsin(tanhx) = arctan(sinhx)?

Answer 1

Please see below.

Explanation:

Let `arcsin(tanhx) =y`

then `siny=tanhx`

Squaring and subtracting it from one, we get

`1-sin^2y=1-tanh^2x`

or `cos^2y=sech^2x`

or `sec^2y=cosh^2x`

or `tan^2y+1=cosh^2x`

or `tan^2y=cosh^2x-1=sinh^2x`

or `tany=sinhx`

or `y=arctan(sinhx)`

Hence `arcsin(tanhx)=arctan(sinhx)`