What is the derivative of, (cos x)cos x

kindly show complete solution,

Following Kasab's ground-breaking insight that cos(x) ͯ cos(x) = cos²(x) which we may conveniently call "f", I venture to proceed, using the product rule for differentiation, to the answer to your question which is:

df/dx = -2.cos(x) ͯ sin(x) = -sin(2x)

I am privileged to stand upon the shoulders of a giant.

Please, clarify what you mean by, (cos x)cos x . If was just the product then it is simpler to write,(cos x)^2. Check!