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.
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