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prove that cos 7x + cos 5x + cos 3x + cos x = 4 cos x cos 2x cos 4x

prove that cos 7x + cos 5x + cos 3x + cos x = 4 cos x cos 2x cos 4x

Grade:11

1 Answers

apurv
49 Points
3 years ago
Rewriting the Left hand side as:
LHS: cos7x + cosx + cos5x + cos3x
Apply the formula : cos a + cos b = 2{cos(a+b)/2}{cos(a-b)/2} on first two terms and other two terms.
LHS: 2{cos(8x)/2}{cos(6x)/2} + 2{cos(8x)/2}{cos(2x)/2}
LHS: 2cos 4x cos 3x + 2 cos 4x cos x
Taking out 2 cos4x common 
LHS: 2cos 4x (cos 3x + cos x)
Now again apply the same  formula
LHS: 2cos 4x ( 2{cos (4x)/2}{cos(2x)/2})
LHS: 2cos 4x (2cos 2x cos x)
LHS: 4cos 4x cos 2x cos x = RHS
HENCE PROVED
 

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