Question

$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\cos 38°\mathrm{cosec}52°}{\tan 18°\tan 35°\tan 60°\tan 72°\tan 55°}.$

Answer 1

$$\begin{gathered} \frac{\cos 38°\mathrm{cosec}52°}{\tan 18°\tan 35°\tan 60°\tan 72°\tan 55°} \\ =\frac{\cos 38°\sec \left(90°-52°\right)}{\cot \left(90°-18°\right)\cot \left(90°-35°\right)\tan 60°\tan 72°\tan 55°} \\ =\frac{\cos 38°\sec 38°}{\cot 72°\cot 55°\tan 60°\tan 72°\tan 55°} \\ =\frac{\cos 38°\times {\displaystyle \frac{1}{\cos 38°}}}{{\displaystyle \frac{1}{\tan 72°}}\times {\displaystyle \frac{1}{\tan 55°}}\times \sqrt{3}\times \tan 72°\times \tan 55°} \\ =\frac{1}{\sqrt{3}} \end{gathered}$$