Evaluate :

$\dfrac{\text{sin 80°}}{\text{cos 10°}}$ + sin 59° sec 31°

Solving,

$\Rightarrow \dfrac{\text{sin 80°}}{\text{cos 10°}} + \text{sin 59° sec 31°} \\[1em] \Rightarrow \dfrac{\text{sin 80°}}{\text{cos 10°}} + \text{sin 59° sec 31°} \\[1em] \Rightarrow \dfrac{\text{sin (90 - 10)°}}{\text{cos 10°}} + \text{sin 59° sec (90 - 59)°}$

By formula,

sin(90° - θ) = cos θ and sec(90° - θ) = cosec θ

$\Rightarrow \dfrac{\text{cos 10°}}{\text{cos 10°}} + \text{sin 59° cosec 59°} \\[1em] \Rightarrow 1 + \text{sin 59°} \times \dfrac{1}{\text{sin 59°}} \\[1em] \Rightarrow 1 + 1 \\[1em] \Rightarrow 2.$

Hence, $\dfrac{\text{sin 80°}}{\text{cos 10°}}$ + sin 59° sec 31° = 2.