Question

Evaluate the following :
$\frac{cos{70}^{\circ }}{sin{20}^{\circ }}+\frac{cos{55}^{\circ }cosec{35}^{\circ }}{tan{5}^{\circ }tan{25}^{\circ }tan{45}^{\circ }tan{65}^{\circ }tan{85}^{\circ }}$

Answer 1

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

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

$=\frac{sin{20}^{\circ }}{sin{20}^{\circ }}+\frac{cos{55}^{\circ }sec{55}^{\circ }}{tan{5}^{\circ }tan{25}^{\circ }tan{45}^{\circ }cot{25}^{\circ }cot{5}^{\circ }}$

$=1+\frac{1}{tan{45}^{\circ }}=1+1=2$

$[\because tan{5}^{\circ }cot{5}^{\circ }=1andtan{25}^{\circ }cot{25}^{\circ }=1,sec{55}^{\circ }cos{55}^{\circ }=1]$