I need help understanding these trig identities:

Where I see this identity I also see it followed up by this identity:

Huh? I recall that:

So cos(b)sin(a) becomes:

Right? but this is NOT equal to the sin(a)cos(b) equation above. Shouldn't it be?

I know that xy=yx so shouldn't cos(b)sin(a) = sin(a)cos(b)?

Where is my logic flawed...