Many thanks for your clear explanation.

I have solved for the other components of the problem:

__1,1 component__
sin(x)sin(z)cos(y) + cos(x)sin(y)

= -sin(x)cos(y) + cos(x)sin(y) = -sin(x-y) according to identity 3

__1,2 component__ (as shown in your post)

-cos(x)sin(z)cos(y) + sin(x)sin(y)

= cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

__2,1 component__
-sin(x)sin(z)sin(y) + cos(x)cos(y)

= sin(x)sin(y) + cos(x)cos(y) = cos(x-y) according to identity 4

__2,2 component__
cos(x)sin(z)sin(y) + sin(x)cos(y)

= -cos(x)sin(y) + sin(x)cos(y) = sin(x-y) according to identity 3

I was a bit unsure about component 1,1 but if any of them are wrong, please let me know

thanks!