Show tan 20 tan40 tan60 tan80=3

tan 20° tan 40° tan 60° tan 80°
= (tan 20° tan 40° tan 80°) {tex}\sqrt {3}{/tex} {tex}\left[\because \tan 60^{\circ}=\sqrt{3}\right]{/tex}
{tex}=\left(\frac{\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}}\right) \sqrt{3}{/tex}
{tex}=\frac{\left(2 \sin 20^{\circ} \sin 40^{\circ}\right) \sin 80^{\circ} \times \sqrt{3}}{\left(2 \cos 20^{\circ} \cos 40^{\circ}\right) \cos 80^{\circ}}{/tex}
Applying
{tex}\Rightarrow{/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{\left(\cos \left(40^{\circ}-20^{\circ}\right)-\cos \left(40^{\circ}+20^{\circ}\right)\right) \sin 80^{\circ} \times \sqrt{3}}{\left(\cos \left(20^{\circ}+40^{\circ}\right)+\cos \left(40^{\circ}-20^{\circ}\right)\right) \cos 80^{\circ}}{/tex}
{tex}=\frac{\left(\cos 20^{\circ}-\cos 60^{\circ}\right) \sin 80^{\circ} \times \sqrt{3}}{\left(\cos 60^{\circ}+\cos 20^{\circ}\right) \cos 80^{\circ}}{/tex}
{tex}=\frac{\left(\cos 20^{\circ}-\frac{1}{2}\right) \sin 80^{\circ} \times \sqrt{3}}{\left(\frac{1}{2}+\cos 20^{\circ}\right) \cos 80^{\circ}}{/tex}
{tex}=\frac{\left(2 \sin 80^{\circ} \cos 20^{\circ}-\sin 80^{\circ}\right) \sqrt{3}}{\cos 80^{\circ}+2 \cos 20^{\circ} \cos 80^{\circ}}{/tex}
{tex}\Rightarrow{/tex} 2 sin A cos B - sin (A + B) + sin (A - B)
{tex}=\frac{\left(\sin \left(80^{\circ}+20^{\circ}\right)+\sin \left(80^{\circ}-20^{\circ} \right)-\sin \theta 0^{\circ}\right) \sqrt{3}}{\cos 80^{\circ}+\left(\cos \left(20^{\circ}+80^{\circ}\right)+\cos \left(80^{\circ}-20^{\circ}\right)\right)}{/tex}
{tex}=\frac{\left(\sin 100^{\circ}+\sin 60^{\circ}-\sin 80^{\circ}\right) \sqrt{3}}{\cos 80^{\circ}+\cos 100^{\circ}+\cos 60^{\circ}}{/tex}
{tex}=\frac{\left(\sin \left(180^{\circ}-80^{\circ}\right)+\frac{\sqrt{3}}{2}-\sin 80^{\circ}\right) \sqrt{3}}{\cos 80^{\circ}+\cos \left(180^{\circ}-80^{\circ}\right)+\cos 60^{\circ}}{/tex}
{tex}=\frac{\left(\sin 80^{\circ}+\frac{\sqrt{3}}{2}-\sin 80^{\circ}\right) \sqrt{3}}{\cos 80^{\circ}-\cos 80^{\circ}+\cos 60^{\circ}}{/tex}
{tex}=\frac{\frac{3}{2}}{\frac{1}{2}}{/tex} = 3 = RHS