Having a bit of difficulty with some calculus where I have to differentiate sin,cos,tan, sinh,cosh and tanh when they are in terms of their exponential functions, in accordance with DeMoivre's Theorem

In particular, d/dt(tan t)

i have gone down this route:

= d/dt(sint/cost) = d/dt {(exp it - exp -it) / i(exp it + exp -it)}

I am unsure about the next step, as it seems everything I have tried gives me a lot of mess!

I have tried multipplying top and bottom by exp it and differentiated to get something in terms of exp 4it

This doesn't help at all!

I know I'm aiming for sec^2 x... so looking for 1/ (exp it + exp -it)

but i can't get there

also finding it difficult to do d/dt (tanh t), but i have successfully completed sin t, cos t, sinh t and cosh t

Please could someone point in the right direction with the question mentioned above? I'm going round in circles!!

Cheers