How to prove tan z is analytic using cauchy-riemann conditions

A function is analytic if the C-R conditions are satisfied:



We have



Recall the angle sum formula for tangent:



Now recall that



So we have







We could stop here, but taking derivatives may be messy and would be easier to do if we can write this in terms of sines and cosines.





With similar usage of identities, we can find that



Now we check the C-R conditions.





Similarly, you can check that , hence the C-R conditions are satisfied, and so is analytic.

What’s your question?