In this exercise well use the accompanying figure to prove

Chapter 8, Problem 56

(choose chapter or problem)

In this exercise well use the accompanying figure to prove the following identities: cos 2u 2 cos2 u 1 sin 2u 2 sin u cos u cos 3u 4 cos3 u 3 cos u sin 3u 3 sin u 4 sin3 u [The figure and the technique in this exercise are adapted from the article by Wayne Dancer, Geometric Proofs of Multiple Angle Formulas, in American Mathematical Monthly, vol. 44 (1937), pp. 366 367.] The figure is constructed as follows. We start with in standard position in the unit circle, as shown. The point C is chosen on the extended diameter such that CA 1. Then is extended to meet the circle at D, and radius is drawn. Finally, from D a perpendicular is drawn to the diameter, as shown. Supply the reason or reasons that justify each of the following statements. (a) (b) Hint: is an exterior angle to ^AOC.(c) (d) (e) From O, draw a perpendicular to meeting at F. From A, draw a perpendicular to meeting at G. Then GC OG cos u and FA DF cos 2u. (f) Hint: Use ^CFO. (g) cos 2u 2 cos2 u 1 Hint: In the equation in part (f), solve for cos 2u. (h) OF sin 2u (i) sin 2u 2 sin u cos u Hint: Find sin u in ^CFO, and then solve the resulting equation for sin 2u. ( j) DC 1 2 cos 2u and EO cos 3u (k) Hint: Compute cos u in ^CDE, and then use part ( j). (l) cos 3u 4 cos3 u 3 cos u Hint: Use the results in parts (k) and (g). (m) DE sin 3u (n) sin 3u 3 sin u 4 sin3 u Hint: Compute sin u in ^CDE.