Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 218: 17

$$\cos40^\circ\cos50^\circ-\sin40^\circ\sin50^\circ=0$$

$$\cos40^\circ\cos50^\circ-\sin40^\circ\sin50^\circ$$ Remember the identity cosine of a sum: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ That means we can shorten the exercise as follows: $$\cos40^\circ\cos50^\circ-\sin40^\circ\sin50^\circ=\cos(40^\circ+50^\circ)$$ $$\cos40^\circ\cos50^\circ-\sin40^\circ\sin50^\circ=\cos90^\circ$$ $$\cos40^\circ\cos50^\circ-\sin40^\circ\sin50^\circ=0$$