The trick you need here is to change every term containing sin^2 to -sinh^2

(after converting it so it contains sin's and cos's only) in a trig identity to get

the corresponding hyperbolic identity.

So

cot^2 +1=cosec^2

becomes:

sin^2/cos^2 + 1 = 1/sin^2.

So the corresponding hyperbolic identity is:

-sinh^2 / cosh^2 + 1 = -1/sinh^2

or:

-tanh^2 + 1 = -cosech^2

or:

tanh^2 -1 = cosech^2

RonL