Tan 20 tan 6o tan 80 …

LHS = tan 20° tan 40° tan 80° {tex}=\frac{\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}}{/tex}
{tex}=\frac{\left(2 \sin 20^{\circ} \sin 40^{\circ}\right) \sin 80^{\circ}}{\left(2 \cos 20^{\circ} \cos 40^{\circ}\right) \cos 80^{\circ}}{/tex} 
{tex}=\frac{\left(\cos 20^{\circ}-\cos 60^{\circ}\right) \sin 80^{\circ}}{\left(\cos 60^{\circ}+\cos 20^{\circ}\right) \cos 80^{\circ}}{/tex} [{tex}\because{/tex} 2 sin a sin b = cos (a-b) - cos (a+b), 2 cos a cos b = cos (a+b) + cos (a-b)]
{tex}=\frac{\sin 80^{\circ} \cos 20^{\circ}-(1 / 2) \sin 80^{\circ}}{(1 / 2) \cos 80^{\circ}+\cos 80^{\circ} \cos 20^{\circ}}{/tex} [ {tex}\because \cos\;60^o=\frac12{/tex}]
{tex}=\frac{2 \sin 80^{\circ} \cos 20^{\circ}-\sin 80^{\circ}}{\cos 80^{\circ}+2 \cos 80^{\circ} \cos 20^{\circ}}{/tex}
{tex}=\frac{\sin 100^{\circ}+\sin 60^{\circ}-\sin 80^{\circ}}{\cos 80^{\circ}+\cos 100^{\circ}+\cos 60^{\circ}}{/tex}[{tex}\because{/tex} 2 sin a cos b = sin (a+b) + sin (a-b)]
{tex}=\frac{\sin \left(180^{\circ}-80^{\circ}\right)+\sin 60^{\circ}-\sin 80^{\circ}}{\cos 80^{\circ}+\cos \left(180^{\circ}-80^{\circ}\right)+\cos 60^{\circ}}{/tex}
{tex}=\frac{\sin 80^{\circ}+\sin 60^{\circ}-\sin 80^{\circ}}{\cos 80^{\circ}-\cos 80^{\circ}+\cos 60^{\circ}}{/tex} {tex}=\frac{\sin 60^{\circ}}{\cos 60^{\circ}}{/tex} = tan 60° = RHS