The value of $$\frac{\tan 13^\circ \tan 37^\circ \tan 45^\circ \tan 53^\circ \tan 77^\circ}{2 \cosec^2 60^\circ(\cos^2 60^\circ - 3 \cos 60^\circ +2) }$$ is:

= $$\frac{\tan 13^\circ \tan 37^\circ \tan 45^\circ \tan 53^\circ \tan 77^\circ}{2 \cosec^2 60^\circ(\cos^2 60^\circ - 3 \cos 60^\circ +2) }$$

Put the trigonometry value.

= $$\frac{\left(\tan13^{\circ}\ \tan77^{\circ}\right)\times\ \left(\tan37^{\circ}\ \tan53^{\circ}\right)\ \times\ 1}{2\times\ \left(\frac{2}{\sqrt{\ 3}}\right)^2\left(\left(\frac{1}{2}\right)^2-3\times\ \frac{1}{2}+2\right)}$$

= $$\frac{\left(\tan13^{\circ}\ \tan\left(90-13\right)^{\circ}\right)\times\ \left(\tan37^{\circ}\ \tan\ \left(90-37\right)^{\circ}\right)\ }{2\times\ \left(\frac{4}{3}\right)\times\ \left(\frac{1}{4}-\frac{3}{2}+2\right)}$$

$$=\frac{\left(\tan13^{\circ}\ \times\ \ \cot13^{\circ}\right)\times\ \left(\tan37^{\circ}\ \ \times\ \cot\ 37^{\circ}\right)\ }{\frac{8}{3}\times\ \left(\frac{1}{4}-\frac{3}{2}+2\right)}$$

= $$\frac{\left(\tan13^{\circ}\ \times\ \frac{1}{\tan13^{\circ}}\right)\times\ \left(\tan37^{\circ}\ \ \times\ \frac{1}{\tan37^{\circ}}\right)\ }{\frac{8}{3}\times\ \left(0.25-1.5+2\right)}$$

= $$\frac{\left(1\times1\right)}{\frac{8}{3}\times0.75}$$

= $$\frac{1}{8\times0.25}$$

= $$\frac{1}{2}$$