Prove that: 36^ 42^ 60^ 78^ =1 16 - Vidyadip

Question

Prove that: $\cos36^\circ\cos42^\circ\cos60^\circ\cos78^\circ=\frac{1}{16}$

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Answer

$\text{LHS}=\cos36^\circ.\cos42^\circ.\cos60^\circ.\cos78^\circ$ $=\frac{1}{2}\cos36^\circ.\cos60^\circ\big(2\cos42^\circ.\cos78^\circ\big)$ $=\frac{1}{2}\bigg(\frac{\sqrt{5}+1}{4}\bigg).\frac{1}{2}(\cos120^\circ+\cos36^\circ)$ $=\frac{\big(\sqrt{5}+1\big)}{16}\Big(\frac{-1}{2}+\frac{\sqrt{5}+1}{4}\Big)$ $=\frac{\sqrt{5}+1}{16}\Big(\frac{-2+\sqrt{5}+1}{4}\Big)$ $=\frac{\big(\sqrt{5}+1\big)\big(\sqrt{5}-1\big)}{64}$ $=\frac{5-1}{64}$ $\frac{1}{16}$ $=\text{RHS}$

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