InterviewSolution
| 1. |
Prove that: cos 78° cos 42° cos 36° = 1/8 |
|
Answer» Let us consider the LHS cos 78° cos 42° cos 36° Now let us multiply and divide by 2 we get, cos 78° cos 42° cos 36° = 1/2 (2 cos 78° cos 42° cos 36°) As we know, 2 cos A cos B = cos (A + B) + cos (A – B) Now the above equation becomes, = 1/2 (cos (78° + 42°) + cos (78° – 42°)) × cos 36° = 1/2 (cos 120° + cos 36°) × cos 36° = 1/2 (cos (180° – 60°) + cos 36°) × cos 36° = 1/2 (-cos (60°) + cos 36°) × cos 36° [since, cos(180° – A) = – A] = 1/2 (-1/2 + (√5 + 1)/4) ((√5 + 1)/4) [since, cos 36° = (√5 + 1)/4] = 1/2 (√5 + 1 – 2)/4 ((√5 + 1)/4) = 1/2 (√5 – 1)/4) ((√5 + 1)/4) = 1/2 ((√5)2 – 12)/16 = 1/2 (5 - 1)/16 = 1/2 (4/16) = 1/8 = RHS Thus proved. |
|