Need to Simplify Trig Functions of ArcTan

Questions
,
1

This works

In[24]:= Cos[ArcTan[x, y]] // TrigReduce
Out[24]= x/Sqrt[x^2 + y^2]

In[25]:= Sin[ArcTan[x, y]] // TrigReduce
Out[25]= y/Sqrt[x^2 + y^2]

but

In[26]:= Cos[3/4 ArcTan[x, y]] // TrigReduce
Out[26]= Cos[3/4 ArcTan[x, y]]

In[52]:= Sin[3/4 ArcTan[x, y]] // TrigReduce
Out[52]= Sin[3/4 ArcTan[x, y]]

How can I get these expressions in terms of x and y ?

2

Maybe:

 {FullSimplify[Cos[3/4 ArcTan[x, y]] // TrigToExp, 
   Assumptions -> {x > 0, y > 0}], 
  FullSimplify[Sin[3/4 ArcTan[x, y]] // TrigToExp, 
    Assumptions -> {x > 0, y > 0}] // Expand}

(*{((x + I y)^(3/2) + (x^2 + y^2)^(3/4))/(
 2 (x + I y)^(3/4) (x^2 + y^2)^(
  3/8)), -((I (x + I y)^(3/4))/(2 (x^2 + y^2)^(3/8))) + (
  I (x^2 + y^2)^(3/8))/(2 (x + I y)^(3/4))}*)
3

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