Properties sine and cosine of complex number

Definition:
The sine and cosine of complex variable is defined as
Sinz=(eiz-e-iz)/2i and cosz=(eiz+e-iz)/2
Properties sine and cosine of complex number:
Properties of sine for complex number:
1-sin(z+2k?)=sinz
Proof:
L.H.S= sin(z+2k?) =(ei(z+2k?)-e-i(z+2k?))/2i =(eiz.ei2k?-e-iz.e-i2k?)/2i
=(eiz-e-iz)/2i = sinz (as e±i2k?=cos2k?±isin2k?=1)
2-sin(z1+z2 )=sinz1cosz2+cosz1sinz2
Proof:
R.H.S=sinz1cosz2+cosz1sinz2
=((eiz1-e-iz1)/2i ).( (eiz2+e-iz2)/2)+( (eiz1+e-iz1)/2).( (eiz2-e-iz2)/2i )
= (ei(z1+z2)+ei(z1-z2)-ei(z2-z1)-e-i(z1+z2)+ei(z1+z2)-ei(z1-z2)+ei(z2-z1)-e-i(z1+z2)) /4i
= (2ei(z1+z2)-2e-i(z1+z2)) /4i
=(ei(z1+z2)-e-i(z1+z2))/2i
= sin(z1+z2 )
3- sin(z1-z2 )=sinz1cosz2-cosz1sinz2
4- sin(z+? )=-sinz
5- sin(z+?/2 )=cosz
6- sinz 2=sin2x+sinh2y
7-sin(-z)=-sinz
8-sin(iz)=isinhz
9-d/dz sinz=cosz
Properties of cosine for complex number:
1-d/dz cosz=-sinz
2-cos(-z)=cosz
3-cos(z1+z2)=cosz1cosz2-sinz1sinz2
4-cos(z1-z2)= cosz1cosz2+sinz1sinz2
5-cos2z=1-sin2z
6-cos(z+?)=-cosz
7-cos2z=cos2z-sin2z
=2cos2z-1
=1-2sin2z
8-cosz1+cosz2=2cos(z1+z2)/2.cos(z1-z2)/2
9- cosz1-cosz2=-2sin(z1+z2)/2.sin(z1-z2)/2
10-cos(iz)=coshz
11-cosz=cosx coshy-isinx sinhy
12- cosz 2=cos2x+sinh2y
13-cosz=0 iff z=(2k+1)?/2 ,k?z
14- cosz = cosx
15-cosz=cosz
16-cos(iz)=cos(iz)