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

To derive the formula for \( \sin(\alpha - \beta) \) using \( \cos(\alpha - \beta) \) and co-function identities, follow these steps: ### 1. Recall the Co-Function Identity The sine and cosine functions are co-functions, meaning: \[ \sin \theta = \cos\left(\frac{\pi}{2} - \theta\right) \] or in degrees, \[ \sin \theta = \cos\left(90^\circ - \theta\right) \] ### 2. Express \( \sin(\alpha - \beta) \) Using the Co-Function Identity Apply the co-function identity to \( \sin(\alpha - \beta) \): \[ \sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - (\alpha - \beta)\right) \] Simplify the argument of the cosine function: \[ \sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - \alpha + \beta\right) \] \[ \sin(\alpha - \beta) = \cos\left(\left(\frac{\pi}{2} - \alpha\right) + \beta\right) \] ### 3. Apply the Cosine of a Sum Formula Use the cosine addition formula: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] Let \( A = \frac{\pi}{2} - \alpha \) and \( B = \beta \). Substituting these into the formula gives: \[ \sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - \alpha\right) \cos \beta - \sin\left(\frac{\pi}{2} - \alpha\right) \sin \beta \] ### 4. Utilize Co-Function Identities to Simplify Apply the co-function identities to the cosine and sine terms: \[ \cos\left(\frac{\pi}{2} - \alpha\right) = \sin \alpha \] \[ \sin\left(\frac{\pi}{2} - \alpha\right) = \cos \alpha \] Substitute these back into the equation: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \] ### 5. Final Formula Thus, the formula for \( \sin(\alpha - \beta) \) is: \[ \boxed{\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta} \] This derivation shows how the sine difference formula can be obtained from the cosine difference formula using co-function identities.