1. Problem: Find the exact values without a calculator for: a) $\cos 40^\circ \cos 50^\circ - \sin 40^\circ \sin 50^\circ$ b) $\sin 40^\circ \cos 20^\circ + \cos 40^\circ \sin 20^\circ$ 2. Formula and rules: Recall the cosine and sine addition formulas: $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$ $$\sin(A+B) = \sin A \cos B + \cos A \sin B$$ These formulas allow us to rewrite the expressions as single trigonometric functions. 3. Solution for a): Using the cosine addition formula: $$\cos 40^\circ \cos 50^\circ - \sin 40^\circ \sin 50^\circ = \cos(40^\circ + 50^\circ) = \cos 90^\circ$$ Since $\cos 90^\circ = 0$, the exact value is 0. 4. Solution for b): Using the sine addition formula: $$\sin 40^\circ \cos 20^\circ + \cos 40^\circ \sin 20^\circ = \sin(40^\circ + 20^\circ) = \sin 60^\circ$$ We know $\sin 60^\circ = \frac{\sqrt{3}}{2}$. Final answers: a) 0 b) $\frac{\sqrt{3}}{2}$