First and second partial derivatives

I previously made the following post:
The problem is: Z = X COS Y - Y COS X

I am to find all of the first and second partial derivatives of this problem.

Will you please solve this problem in detail so that I may use the method you show to solve similar problems?

I received the following response:

The problem is: Z = X COS Y - Y COS X

Solution. As Z = X COS Y - Y COS X,

                       Zx=cosY+YsinX and Zy=-XsinY-cosX

So,
                       Zxx=YcosX,

                       Zyy=-XcosY
and
                       Zxy=-sinY+sinX

Note: Zx and Zy denote the first partial derivatives with respect to X and Y, respectively. Similarly, Zxx, Zyy and Zxy denote the second partial derivatives.

I guess that I forgot to add to my post that I already had the answers and was really looking for how the answers were arrived at.

Will someone please give me a detailed, step by step, description of how these answers were found?

Thank you