Prove that: 6 42^ 66^ 78^ =1 16 - Vidyadip

Question

Prove that:
$\sin6\circ\sin42^\circ\sin66^\circ\sin78^\circ=\frac{1}{16}$

✓

Answer

$\text{LHS}=\cos6^\circ\cos42^\circ\cos66^\circ\cos78^\circ$

$=\frac{1}{4}(2\cos6^\circ\cos66^\circ)(2\cos42^\circ\cos78^\circ)$

$=\frac{1}{4}(\cos72^\circ+\cos60^\circ)(\cos120^\circ+\cos36^\circ)$

$=\frac{1}{4}\Big(\sin18^\circ+\frac{1}{2}\Big)\bigg(-\frac{2}{2}+\frac{\sqrt{5}+1}{4}\bigg)$

$=\frac{1}{4}\bigg(\frac{\sqrt{5}-1}{4}+\frac{1}{2}\bigg)\bigg(\frac{\sqrt{5}+1}{4}-\frac{1}{2}\bigg)$

$=\frac{1}{4}\bigg(\frac{\sqrt{5}-1+2}{4}\bigg)\bigg(\frac{\sqrt{5}+1-2}{4}\bigg)$

$=\frac{1}{64}(\sqrt{5}+1)(\sqrt{5}-1)$

$=\frac{1}{64}(\sqrt{5})^2-1^2)$

$=\frac{1}{64}(5-1)$

$=\frac{1}{16}$

$=\text{RHS}$

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