Z transform examples

Find Z transform of F[K] =Cos(ak) u[k] :

F(z) = Z (cos(ak) u(k) )

= Z ( ½ (e^jak+e^-jak) u(K) )

= ½ (Z (e^jak  u(k) ) + Z (e^-jak) u(K) )

=½ ( Z/ Z-e^ja + Z/ Z-e^-ja)

= ½ ( Z/ Z-e^ja * Z-e^-ja/ Z-e^-ja + Z/ Z-e^-ja * Z-e^-ja/ Z-e^-ja )

=½ ( Z²- Ze^-ja / Z²- Ze^ja – Ze^-ja +1  + Z²- Ze^ja / Z²- Ze^ja – Ze^-ja +1)

=½ (2Z² – (Ze^ja – Ze^-ja) / Z²- Ze^ja – Ze^-ja +1)

= ½(2Z² – 2Zcos(a)/ Z²-2Zcos(a) +1 

= Z (Z-cos(a)) / Z²-2Zcos(a) +1