# Tan 13tan 37 tan 45 tam 53 tan 770.2 Co sec? 60Â°Clasa 60Â°-3.605 60Â°+2)?

$\frac{\tan 13^{\circ} \times \tan 37^{\circ}\times \tan45^{\circ}\times \tan53^{\circ}\times \tan77^{\circ}}{2\times \csc ^2 60^{\circ}(\cos^2 60^{\circ}-3\times \cos 60^{\circ}+2)}\\*We\;know\;that\;\tan45^{\circ}=1,\; \cos60^{\circ}=\frac{1}{2},\; \csc60^{\circ}=\frac{\sqrt{3}}{2}\\*Now,\; first\; solve\;for\;numerator\;-\\*\tan 13^{\circ} \times \tan 37^{\circ}\times \tan45^{\circ}\times \tan53^{\circ}\times \tan77^{\circ}\\* \because \tan(90-\theta )=\cot\theta\; and\;\tan\theta\times \cot\theta=1,\\* \tan(90-13)^{\circ} \times \tan(90-37)^{\circ}\times \tan45^{\circ} \times \tan53^{\circ}\times \tan77^{\circ}\\* \because \cot77^{\circ}\times \tan77^{\circ}=cot53^{\circ}\times tan53^{\circ}=1\; and \;also\; \tan45^{\circ}=1\\* Now,\; solve\;for\;denominator \;-\;As \;we\;know\;the\;values\;\\*We\;will\;directly\;put\;values,\;we\;get,\\* 2\times \csc^260^{\circ}\times (\cos^2 60^{\circ}-3\times \cos60^{\circ}+2) \\*\frac{9}{8}\\*Hence, \frac{1}{\frac{9}{8}}=\frac{8}{9}$

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