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 $arcsin(tanhx)=arctan(sinhx)$