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Prove that: \(4\cos 12^{\circ }\cos 48^{\circ }\cos 72^{\circ }=\cos 36^{\circ}\)
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Prove that: \(4\cos 12^{\circ }\cos 48^{\circ }\cos 72^{\circ }=\cos 36^{\circ}\)

Answer

LHS \(=4 \cos 12^{\circ} \cos 48^{\circ} \cos 72^{\circ}\)
\(\Rightarrow \quad \mathrm{LHS}=2\left(2 \cos 12^{\circ} \cos 48^{\circ}\right) \cos 72^{\circ}\)
\(\Rightarrow \quad \text{LHS} =2\left(\cos 60^{\circ}+\cos 36^{\circ}\right) \cos 72^{\circ}\)
\(\Rightarrow \quad \text{LHS} =2 \cos 60^{\circ} \cos 72^{\circ}+2 \cos 36^{\circ} \cos 72^{\circ}\)
\(\Rightarrow \quad \mathrm{LHS}=\cos 72^{\circ}+\cos 108^{\circ}+\cos 36^{\circ}\)
\(\Rightarrow \quad \text{LHS}=\cos 72^{\circ}+\cos \left(180^{\circ}-72^{\circ}\right)+\cos 36^{\circ}=\cos 72^{\circ}-\cos 72^{\circ}+\cos 36^{\circ}=\cos 36^{\circ}=\mathrm{RHS}\)
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