Question

Evaluate : \(\cos 12 \cos 24 \cos 36 \cos 48 \cos 72 \cos 84\)

Answer 1

The answer in itself is 1/128 and here is the procedure to prove it:
cos(A)*cos(60+A)*cos(60-A) = cos(A)*(cos²60 - sin²A) 

= cos(A)*{(1/4) - 1 + cos²A} = cos(A)*(cos²A - 3/4) 

= (1/4){4cos^3(A) - 3cos(A)} = (1/4)*cos(3A) 

Now we group applying what we see above

cos(12)*cos(48)*cos(72) = 
=cos(12)*cos(60-12)*cos(60+12) = (1/4)cos(36) 

Similarly, cos(24)*cos(36)*cos(84) = (1/4)cos(72) 

Now the given expression is: 

= (1/4)cos(36)*(1/4)*cos(72)*cos(60) = 

= (1/16)*(1/2)*{(√5 + 1)/4}*{(√5 - 1)/4} [cos(60) = 1/2; 
cos(36) = (√5 + 1)/4 and cos(72) = cos(90-18) = 
= sin(18) = (√5 - 1)/4] 
And we seimplify it and it goes: (1/512)*(5-1) = 1/128