Question

Without using trigonometric tables, evaluate the following:
$\frac{\sec {37}^o}{co\sec {53}^o}+2\cot {15}^o\cot {25}^o\cot {45}^o\cot {75}^o\cot {65}^o-3({\sin }^2{18}^o+{\sin }^2{72}^o)$

Answer 1

$\text{sec}x=\text{cosec}(\frac{\pi }{2}-x);\text{cot}x=\tan (\frac{\pi }{2}-x);\cos x=\sin (\frac{\pi }{2}-x)$

$\frac{\text{sec}{37}^{\circ }}{\text{sec}{37}^{\circ }}+2\tan {75}^{\circ }\tan {65}^{\circ }\text{cot}{75}^{\circ }\text{cot}{65}^{\circ }-3({\sin }^2{18}^{\circ }+{\cos }^2{18}^{\circ })=1+2-3=0$