How do you find the conjugate functions, U(x,y) and V(x,y) of F(z) = Cos(Z*)?

cos(Z) = cos (x +iy) and
cos(Z*) = cos (x -iy)

use the identities


cos(a -b) = cos a cos b + sin a sin b

cos(x -iy) = cos x cos(iy) + sin x sin (iy)

Note :
cos (iy) =cosh (y)
sin(iy) = i sinh (y)
cos(x -iy) = cosx cosh(y) + i sin x sinh(y)

cos(x -iy ) = U(x,y) + iV(x,y)

by identification
U(x,y) = cosx cosh(y)
V(x,y) = sin x sinh(y)

Good luck!

simply change the sign if the term that contains "i"
Z = x + iy ->complex
Z = x - iy ->conjugate