What do you notice concerning the values of \( \cos (\alpha-\beta) \) and \( \cos \alpha \) (Hint - are the values the same or different?)

When observing the values of \( \cos (\alpha - \beta) \) and \( \cos \alpha \), it’s important to recognize that they are generally different due to the nature of the cosine function and the angle subtraction involved. The equation \( \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \) shows that \( \cos (\alpha - \beta) \) is a combination of both sine and cosine terms, thus making it dependent on both angles, while \( \cos \alpha \) only relies on the angle \( \alpha \). In specific scenarios, like when \( \beta = 0\), the values align perfectly as \( \cos (\alpha - 0) = \cos \alpha \). However, for most other angles, particularly when \( \beta \) is not zero, we’ll see diverging behaviors which reflect the unique influences of both angles on the cosine function. So, you’re in for a rollercoaster ride of trigonometric fun!