SOLUTION: To prove:-
cos12.cos24.cos48.cos84=1/16
can anyone help me with this................
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Question 466502: To prove:-
cos12.cos24.cos48.cos84=1/16
can anyone help me with this................
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
First we should rewrite cos 84 as sin 6 (these two are equal). Denote our desired answer A (which should turn out to be 1/16)
(\cos 12)(\cos 24)(\cos 48) = A)
(remember sin 6 is the cos 84)
Suppose we multiply both sides by 16 cos 6:
(\cos 6)(\cos 12)( \cos 24 )(\cos 48) = 16 (\cos 6)A)
Applying the double angle identity for sine, we have 2 sin 6 cos 6 = sin 12, and we replace sin 6 cos 6 with (1/2) sin 12 (here I will divide coefficient by 2)
(\cos 12)(\cos 24)(\cos 48) = 16 (\cos 6)A)
We apply it again and again:
(\cos 24)(\cos 48) = 16 (\cos 6)A)
(\cos 48) = 16(\cos 6)A)
A)
Here, sin 96 is equal to cos 6 (you can verify this either using the sum identity or the unit circle), so we can cancel them out, leaving
, as desired.
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