Question

Prove by C/D formulas that cos 6°×cos66°×cos42°×cos78° = 1/16​

Answer 1

True.

Explanation:

(LHS) cos 6∘ cos 42∘ cos 66∘ cos 78∘

1/4 [cos(66∘+6∘) = cos(66∘-6∘)] • [cos(78∘+42∘)+cos(78∘-42∘)]

= 1/4(sin18∘+12)(-12+cos36∘)

[∵cos72∘ = cos(90∘−18∘) = sin18∘]

1/4[(5-√-1)/4+1/2][−1/2+(5-√+1)/4]

[∵sin(18∘ = (5-√-1)/4, and, cos 36∘ = (5-√+1)/4]

1/4(5-√+1)/4(5-√-1)/4 = (5-1)/64

= 4/64 = 1/16 (RHS)

Therefore, the cosine is equal to 1/16.