The value of 3 cosec 20^ - 20^ 20^ 40^ 60^ 80^ is equal to

Numerator:
$\sqrt{3}\operatorname{cosec} 20° - \sec 20° = \frac{\sqrt{3}\cos 20° - \sin 20°}{\sin 20° \cos 20°}$
$\sqrt{3}\cos 20° - \sin 20° = 2$\left(\frac{\sqrt{3}}{2}\cos 20° - \frac{1}{2}\sin 20°\right)$<br>$= 2\cos(20° + 30°) = 2\cos 50°$<br>So numerator =$\frac{2\cos 50°}{\sin 20° \cos 20°} = \frac{4\cos 50°}{\sin 40°} = 4$<br>Denominator:<br>$\cos 20° \cos 40° \cos 60° \cos 80°$where$\cos 60° = \frac{1}{2}$<br>Using product formula:<br>$\sin 20° \cos 20° \cos 40° \cos 80° = \frac{1}{8}\sin 20°$<br>Therefore$\cos 20° \cos 40° \cos 80° \cdot \frac{1}{2} = \frac{1}{16}$<br>So,$\frac{4 \cdot \sin 20° \cos 20°}{\frac{1}{16}} = 64