2.15 Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha-\beta) \). (Hlint; Use co-function)

To derive a formula for \( \sin(\alpha - \beta) \) using \( \cos(\alpha - \beta) \), we start with the co-function identity \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \). Applying this, we have: \[ \sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - (\alpha - \beta)\right) = \cos\left(\frac{\pi}{2} - \alpha + \beta\right). \] Now we can use the cosine angle difference formula: \[ \cos\left(\frac{\pi}{2} - \alpha + \beta\right) = \cos\left(\frac{\pi}{2} - \alpha\right)\cos(\beta) - \sin\left(\frac{\pi}{2} - \alpha\right)\sin(\beta). \] Using co-function identities: - \( \cos\left(\frac{\pi}{2} - \alpha\right) = \sin(\alpha) \) - \( \sin\left(\frac{\pi}{2} - \alpha\right) = \cos(\alpha) \) We substitute these into the formula: \[ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta). \] And voilà! We have derived the formula: \[ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta). \]