Mathematics

Problem

Starting only with the standard addition formulas for sine and cosine, (a) derive an identity for

\sin A+\sin B

that involves the half–sum and half–difference of

A

and

B

; (b) derive a corresponding identity for

\cos A+\cos B

; (c) using the results obtained, simplify

\sin 54^{\circ}+\sin 6^{\circ}+\cos 54^{\circ}+\cos 6^{\circ}

to a single numerical value. a. Derive an identity for

\sin A+\sin B

that involves the half–sum and half–difference of

A

and

B

. b. Derive a corresponding identity for

\cos A+\cos B

. c. Using the results obtained, simplify

\sin 54^{\circ}+\sin 6^{\circ}+\cos 54^{\circ}+\cos 6^{\circ}

to a single numerical value.