Evaluate:

/ (/cos 12 / cos 24 / cos 36 / cos 48 / cos 72 / cos 84/)

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