Question

Evaluate : $\frac{\cos {70}^o}{\sin {20}^o}+\frac{\csc {55}^o.\cos {35}^o}{\tan 5^o.\tan {25}^o.\tan {45}^o.\tan {65}^o.\tan {85}^o}$

Answer 1

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

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

$=\frac{\sin \left({90}^{\circ }-{20}^{\circ }\right)}{\sin {20}^{\circ }}+\frac{\csc {55}^{\circ }\cos {35}^{\circ }}{\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\tan \left(90-{25}^{\circ }\right)\tan \left({90}^{\circ }-5^{\circ }\right)}$

$=\frac{\sin {20}^{\circ }}{\sin {20}^{\circ }}+\frac{\cos \left({90}^{\circ }-{55}^{\circ }\right)\csc {55}^{\circ }}{\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\cot {25}^{\circ }\cot 5^{\circ }}$

$=1+\frac{\sin {55}^{\circ }\csc {55}^{\circ }}{1}$

$=1+1$

$=2$