Answer
The given statement
$$\cos140^\circ=\cos60^\circ\cos80^\circ-\sin60^\circ\sin80^\circ$$
is true.
Work Step by Step
$$\cos140^\circ=\cos60^\circ\cos80^\circ-\sin60^\circ\sin80^\circ$$
As $140^\circ=60^\circ+80^\circ$,
$$\cos140^\circ=\cos(60^\circ+80^\circ)$$
Now we use the cosine sum identity
$$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ (be extremely careful about the sign in the middle)
to expand $\cos(60^\circ+80^\circ)$:
$$\cos140^\circ=\cos60^\circ\cos80^\circ-\sin60^\circ\sin80^\circ$$
This means that the statement
$$\cos140^\circ=\cos60^\circ\cos80^\circ-\sin60^\circ\sin80^\circ$$
is true.
