Prove the following identities: 28. (sin 2x sin 2 y) tan (x + y) = 2 (sin x - sin3y). 29. sin 3x4 sin x sin (60° + x) sin (60°- x). 30. sin 4 x = 2 sin x cos 3x + sin 2 x. 31. sin x + sin(x − π) + sin (π 32. cos x sin (y x) = 0. x) + cos z sin (x − y) = 0. cos (x + z) sin z sin(x + y) cos y sin (x + z) cos z. 34. cos(x + y + z) + cos(x + y) + cos(x - y + z) 35. sin(x+y) cos (xy) + sin (y + z) cos (y-z) +sin(2x) cos (≈ - x)= sin 2x + sin 2 y + sin 2 z. 36. sin(x + y) + cos(x − y) = 2 sin (x + π) sin (y + ‡π). 40. sin x + 2 sin 3x + sin 5 x 4 cos2x sin 3x. = If A, B, C are the angles of a triangle, prove that: 41. sin 2A + sin 2B + sin 2 C = 4 sin A sin B sin C. 42. cos 2A + cos 2B + cos 2 C - 1. 4 cos A cos B cos C'. 43. sin 3A + sin 3B + sin 3 C: 44. cos2A+ cos2 B + cos2 C=1 = If A + B + C = 90°, prove that: 4 cos 3 A cos B cos 3 C. 2 cos A cos B cos C. 45. tan A tan B + tan B tan C+ tan C tan A = 1. 46. sin2A + sin2 B + sin2C = 1 − 2 sin A sin B sin C. 47. sin 2 A + sin 2B + sin 2 C = 4 cos A cos B còs C. 48. Prove that cot-1 3 + csc-1 √5 = 4 π. 49. Prove that x + tan-1 (cot 2x) = tan-1 (cot x). Prove the following statements: 54. tan 11° 15' + 2 tan 22° 30' + 4 tan 45° = cot 11° 15'. 149. How to solve a Trigonometric Equation. To solve a trigonometric equation is to find for the unknown quantity the general value which satisfies the equation. Practically it suffices to find the values between 0° and 360°, since we can then apply our knowledge of the periodicity of the various functions to give us the other values if we need them. There is no general method applicable to all cases, but the following suggestions will prove of value : 1. If functions of the sum or difference of two angles are involved, reduce such functions to functions of a single angle. Thus, instead of leaving sin (x + y) in an equation, substitute for sin (x + y) its equal sin x cos y + cos x sin y. Similarly, replace cos 2 x by cos2 x — sin2x, and replace the functions of by the functions of x. 2. If several functions are involved, reduce them to the same function. This is not always convenient, but it is frequently possible to reduce the equation so as to involve only sines and cosines, or tangents and cotangents, after which the solution can be seen. 3. If possible, employ the method of factoring in solving the final equation. 4. Check the results by substituting in the given equation. For example, solve the equation cos x = sin 2x. By § 101, sin 2x = 2 sin x cos x. .. cos x 2 sin x cos x. .. (1— 2 sin x) cos x = 0. .. cos x = 0, or 1 2 sin x = 0. .. x = 90° or 270°, 30° or 150°, or these values increased by 2 n. Each of these values satisfies the given equation. |