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

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Answer 1 · 2017-11-11T05:25:30

Please see below.

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)#