Question
Sat July 28, 2012 By: Sivendu Sivendu
 

Prove that cos6°cos42°cos66°cos78°=1/16

Expert Reply
Sat July 28, 2012
Answer : Given :  cos6 cos42 cos66 cos78 =1/16
 to prove the given statement .
 
LHS 
cos6 cos42 cos66 cos78 
 Multiplying the dividing by 4  and applying 2cosa cosb= cos(a+b) + cos(a-b) , we get 

= (cos 6) (cos 42)(cos 66) (cos 78)
= (1/4) * [2cos6 cos66] * [2cos42 cos78]
= (1/4) * (cos72 + cos60) * (cos120 + cos36)
= (1/4) (cos72 + 1/2) * (- 1/2 + cos36)
= (1/4) [- 1/4 + (1/2) (cos36 - cos72) + cos36cos72]
= (1/4) [- 1/4 + sin54sin18 + sin54 sin18]
= (1/4) [- 1/4 + 2sin54 sin18 cos18 / cos18]
= (1/4) [- 1/4 + 2sin54 sin36 / cos18]
= (1/4) [- 1/4 + (cos18 - cos90) / 2cos18]
= (1/4) [- 1/4 + 1/2]
= 1/16
= RHS
Hence Proved 
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