Evaluate:

$\dfrac{\text{cos 70°}}{\text{sin 20°}} + \dfrac{\text{cos 59°}}{\text{sin 31°}} - \text{8 sin}^2 30°$

Solving,

$\Rightarrow \dfrac{\text{cos 70°}}{\text{sin 20°}} + \dfrac{\text{cos 59°}}{\text{sin 31°}} - \text{8 sin}^2 30° \\[1em] \Rightarrow \dfrac{\text{cos 70°}}{\text{sin (90° - 70°)}} + \dfrac{\text{cos 59°}}{\text{sin (90° - 59°)}} - \text{8 sin}^2 30°$

By formula,

sin (90° - θ) = cos θ.

$\Rightarrow \dfrac{\text{cos 70°}}{\text{cos 70°}} + \dfrac{\text{cos 59°}}{\text{cos 59°}} - 8 \times \Big(\dfrac{1}{2}\Big)^2 \\[1em] \Rightarrow 1 + 1 - 8\times \dfrac{1}{4} \\[1em] \Rightarrow 2 - 2 \\[1em] \Rightarrow 0.$

Hence, $\dfrac{\text{cos 70°}}{\text{sin 20°}} + \dfrac{\text{cos 59°}}{\text{sin 31°}} - \text{8 sin}^2 30°$ = 0.