Question

Find the value of $\frac{cos{38}^{o}cosec{52}^o}{tan{18}^{o}tan{35}^{o}tan{60}^{o}tan{72}^{o}tan{55}^o}$

Answer 1

$\frac{cos{38}^{o}cosec{52}^o}{tan{18}^{o}tan{35}^{o}tan{60}^{o}tan{72}^{o}tan{55}^o}=\frac{cos{38}^{o}cosec({90}^o-{38}^o)}{tan{18}^{o}tan{35}^o\times \sqrt{3}\times tan({90}^o-{18}^o)tan({90}^o-{35}^o)}\because \left[cosec\right({90}^o-\theta )=sec\theta andtan({90}^o-\theta )=cot\theta ]=\frac{cos{38}^{o}sec-{38}^o}{tan{18}^{o}tan{35}^o\times \sqrt{3}\times cot{18}^{o}cot{35}^o}=\frac{\frac{1}{sec-{38}^o}\times sec-{38}^o}{\frac{1}{cot{18}^o}\times \frac{1}{cot{35}^o}\times \sqrt{3}\times cot{18}^{o}cot{35}^o}=\frac{1}{\sqrt{3}}$