Thread: Can somebody prove - Hyperbolic functions

Could somebody please prove these identities and explain a little bit ? I just can't figure them out.

sinh^-1x = ln(x + sq.rt of(x^2 + 1))

cosh^-1x = ln(x + sq.rt of (x^2 + 1))

tanh^-1x = (1/2)ln ( (1+x)/(1 - x) )

Originally Posted by yakkow

Could somebody please prove these identities and explain a little bit ? I just can't figure them out.

sinh^-1x = ln(x + sq.rt of(x^2 + 1))

I prove just one, you prove the rest.
Since,
is a bijective function it has an inverse.
Thus you interchange the x and y in,

To get,

Thus,

Thus,

Thus,

Allow, thus,

Thus,

Thus,

Which is,

Since, we have,

Thus,

Thus,