Question

Solve:
$\frac{\cos 70}{\sin 20}+\cos 59\csc 31$

Answer 1

$\frac{\cos {70}^{\circ }}{\sin {20}^{\circ }}+\cos {59}^{\circ }\csc {31}^{\circ }$

$=\frac{\cos ({90}^{\circ }-{20}^{\circ })}{\sin {20}^{\circ }}+\cos ({90}^{\circ }-{31}^{\circ })\csc {31}^{\circ }$

We know that $\cos (90-\theta )=\sin \theta$

$=\frac{\sin {20}^{\circ }}{\sin {20}^{\circ }}+\sin {31}^{\circ }\csc {31}^{\circ }$

$=1+\sin {31}^{\circ }\times \frac{1}{\sin {31}^{\circ }}$ since $\csc {31}^{\circ }=\frac{1}{\sin {31}^{\circ }}$

$=1+1=2$