please verify this equation:

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

You could use the compound angle formula:

cos(A + B) = cosAcosB - sinAsinB

Let A = x+y and B = z

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

Use the same identity again:

sin(A+B) = cosAsinB + sinAcosB

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

I'll let you expand and see by yourself :)