The conversation between human and AI assistant.
Why is [math]\cos\alpha \cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)][/math]?
Can you give me an example of how to use the identity?
Sure! Let's say we want to find [math]\cos 40^\circ \cos 50^\circ[/math]. Using the formula [math]\cos x\cos y=\frac{1}{2}[\cos(x-y)+\cos(x+y)][/math], we can write [math]\cos 40^\circ \cos 50^\circ=\frac{1}{2}[\cos(10^\circ)+\cos(90^\circ)][/math]. Simplifying, we get [math]\frac{1}{2}\cdot\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}[/math].