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16. Prove that Cos 20° Cos 40° Cos 60° Cos 80º =116 |
| Answer» COS 20° · Cos40° · Cos60° . Cos80° = 1/16 LHS = Cos 20° · Cos40° · Cos60° . Cos80° We know that Cos60° = 1/2 LHS = Cos 20° · Cos40° · 1/2 . Cos80° Multiply and divide the equation by 2 LHS = 1/4 (2. Cos 20° · Cos40° · Cos80°) We know the formula 2 COSA cosb= cos(a+b) + cos(a-b) LHS = 1/4 [Cos(20+80)+ Cos(20-80)] . Cos40 LHS = 1/4 [Cos(-60)+ Cos(100)] Cos40 LHS = 1/4 [1/2 + cos100] Cos40 LHS = 1/8 Cos40+ 1/4 (Cos40 . Cos100) Multiply and divide the equation by 2 LHS = 2/2 (1/8 Cos 40) + 1/8(2 Cos40 Cos100) We know the formula 2cosa cosb= cos(a+b) cos(a-b) LHS = 1/8 Cos40+ 1/8 [Cos 140 + Cos (-60)] LHS = 1/8 Cos 40+ 1/8 Cos 140 + 1/16 Since Cos 60= 1/2 LHS = 1/8 (Cos 40 + Cos 140) + 1/16 LHS = 1/8 [2 Cos 90 Cos (-50)] + 1/16 LHS = Cos 90 Cos 90 = 0 LHS = 1/16 = RHS HENCE PROVED. | |