Question

Evaluate $\frac{cos{70}^{\circ }}{sin{20}^{\circ }}+\frac{cos{59}^{\circ }}{sin{31}^{\circ }}-8si{n}^2{30}^{\circ }=0$

Answer 1

L.H.S. = $\frac{cos{70}^{\circ }}{sin{20}^{\circ }}+\frac{cos{59}^{\circ }}{sin{31}^{\circ }}-8si{n}^2{30}^{\circ }=0$

$\Rightarrow$ L.H.S. = $\frac{cos({90}^{\circ }-{20}^{\circ })}{sin{20}^{\circ }}+\frac{cos({90}^{\circ }-{31}^{\circ })}{sin{31}^{\circ }}-8si{n}^2{30}^{\circ }$

$\Rightarrow$ L.H.S. = $\frac{sin{20}^{\circ }}{sin{20}^{\circ }}+\frac{sin{31}^{\circ }}{sin{31}^{\circ }}-8si{n}^2{30}^{\circ }$ $[\because cos({90}^{\circ }-\theta )=sin\theta ]$

$\Rightarrow =1+1-8{\left(\frac{1}{2}\right)}^2=2-2=0=R.H.S.$ $[\because sin{30}^{\circ }=\frac{1}{2}]$