Chapter 7.2, Problem 77E

PROVE: Addition Formula for Sine In the text we proved only the Addition and Subtraction Formulas for Cosine. Use these formulas and the cofunction identities

$\begin{array}{l}\sin x=\cos \left(\frac{\pi }{2}-x\right)\hfill \\ \cos x=\sin \left(\frac{\pi }{2}-x\right)\hfill \end{array}$

to prove the Addition Formula for Sine. [Hint: To get started, use the first cofunction identity to write

$\begin{array}{lll}\sin (s+t)\hfill & =\hfill & \cos \left(\frac{\pi }{2}-(s+t)\right)\hfill \\ \hfill & =\hfill & \cos \left(\left(\frac{\pi }{2}-s\right)-t\right)\hfill \end{array}$

and use the Subtraction Formula for Cosine.]