2.10 Ure 29 to show that \( \sin (\alpha-\beta)=\sin \alpha \cos \beta-\sin \beta \cos \alpha \)

To prove that \( \sin (\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha \), we can use the unit circle approach. On the unit circle, the coordinates of points corresponding to angles \( \alpha \) and \( \beta \) can be represented by \( (\cos \alpha, \sin \alpha) \) and \( (\cos \beta, \sin \beta) \). The sine of the difference of two angles can then be visualized using the geometry of the circle and the projection of these points along the y-axis, leading us to the desired sine subtraction identity. Alternatively, we can utilize the angle addition formulas and the definition of sine in terms of the angles involved. By manipulating the addition formulas for sine and cosine, and applying known algebraic identities, we can systematically derive the expression through careful substitution. This approach highlights the interconnectedness of trigonometric identities and enhances understanding of their applications in various mathematical contexts.