This exercise is adapted from a problem that appears in

Chapter 6, Problem 58

(choose chapter or problem)

This exercise is adapted from a problem that appears in the classic text A Treatise on Plane and Advanced Trigonometry, 7th ed., by E. W. Hobson (New York: Dover Publications, 1928). (The first edition of the book was published by Cambridge University Press in 1891.) Given: A, B, and C are acute angles such that cos tan B cos tan C cos C tan A Prove: sin sin B sin C 2 sin 18 Follow steps (a) through (e) to obtain this result. (a) In each of the three given equations, use the identity so that the equations contain only sines and cosines. (b) In each of the three equations obtained in part (a), square both sides. Then use the identity so that each equation contains only the cosine function. (c) For ease in writing, replace cos2 A, cos2 B, and cos2 C by a, b, and c, respectively. Now you have a system of three equations in the three unknowns a, b, and c. Solve for a, b, and c. (d) Using the results in part (c), show that (e) From Exercise 54(f) we know that sin 18 Show that the expression obtained in part (d) is equal to twice this expression for sin 18. This completes the proof. (Use the fact that two nonnegative quantities are equal if and only if their squares are equal.)