Thanks, however, need all four quadrants, but this works.
In[4]:= FullSimplify[Cos[3/4 ArcTan[x, y]] // TrigToExp,
Assumptions -> Element[{x, y}, Reals]]
Out[4]= ((x + I y)^(3/2) + (x^2 + y^2)^(3/4))/(2 (x + I y)^(
3/4) (x^2 + y^2)^(3/8))
In[5]:= FullSimplify[Sin[3/4 ArcTan[x, y]] // TrigToExp,
Assumptions -> Element[{x, y}, Reals]]
Out[5]= -((
I (x^2 + y^2)^(3/8) (-1 + (x + I y)^(3/2)/(x^2 + y^2)^(3/4)))/(
2 (x + I y)^(3/4)))
Also, I'd like to have expressions without the square root of -1