Question

Find the value of the expression
$\frac{2\cos {70}^o}{\sin {20}^o}+\frac{\cos {55}^{o}cose{35}^o}{\tan 5^o\tan {25}^o\tan {45}^o\tan {65}^o\tan {85}^o}$

Answer 1

$\frac{2\cos {70}^{\circ }}{\sin {20}^{\circ }}+\frac{\cos {55}^{\circ }\csc {35}^{\circ }}{\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\tan {65}^{\circ }\tan {85}^{\circ }}$

$=\frac{2\cos {70}^{\circ }}{\sin ({90}^{\circ }-{70}^{\circ })}+\frac{\cos {55}^{\circ }\csc {35}^{\circ }}{\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\tan ({90}^{\circ }-{25}^{\circ })\tan ({90}^{\circ }-5^{\circ })}$

$=\frac{2\cos {70}^{\circ }}{\cos {70}^{\circ }}+\frac{\cos {55}^{\circ }\csc {35}^{\circ }}{\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\cot 5^{\circ }\cot {25}^{\circ }}$

$=2+\cos ({90}^{\circ }-{35}^{\circ })\csc {35}^{\circ }$ since $\tan \theta \cot \theta =1$

$=2+\sec {35}^{\circ }\csc {35}^{\circ }$