\displaystyle}\sin^2z=\sin^2\left( z_0 + z-z_0\right)\\
=\left(\sin z_0\cos w+\cos z_0\sin w\right)^2\\
=\sin^2z_0\cos^2w+2\sin z_0\cos z_0\cos w \sin w +\cos^2z_0\sin^2w\\
=\sin^2z_0+\left(\cos^2z_0-\sin^2z_0\right)\sin^2w+\frac{1}{2}\sin 2z_0\sin 2w