Prove that: 78^ 42^ 36^ =1 8 - Vidyadip

Question

Prove that: $\cos78^\circ\cos42^\circ\cos36^\circ=\frac{1}{8}$

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Answer

$\text{LHS}=\cos78^\circ.\cos42^\circ.\cos36^\circ$ $=\frac{(2\cos78^\circ.\cos42^\circ)}{2}.\cos36^\circ$ $=\frac{1}{2}(\cos120^\circ+\cos36^\circ).\cos36^\circ$ $=\frac{1}{2}\Big(\frac{-1}{2}+\frac{\sqrt{5}+1}{4}\Big)\frac{\sqrt{5}+1}{4}$ $=\frac{1}{8}\frac{\big[-2\big(\sqrt{5}+1\big)+5+1+2\sqrt{5}\big]}{4}$ $=\frac{1}{4}\big[\frac{4}{4}\big]$ $=\frac{1}{8}$ $=\text{RHS}$

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