Evaluate :

$2\dfrac{\text{tan 57°}}{\text{cot 33°}} - \dfrac{\text{cot 70°}}{\text{tan 20°}} - \sqrt{2}$cos 45°

Solving,

$\Rightarrow 2\dfrac{\text{tan 57°}}{\text{cot 33°}} - \dfrac{\text{cot 70°}}{\text{tan 20°}} - \sqrt{2}\text{cos 45°} \\[1em] \Rightarrow 2\dfrac{\text{tan 57°}}{\text{cot (90° - 57°)}} - \dfrac{\text{cot 70°}}{\text{tan (90° - 70°)}} - \sqrt{2}\times \dfrac{1}{\sqrt{2}}$

By formula,

tan(90° - θ) = cot θ and cot(90° - θ) = tan θ

$\Rightarrow 2 \times \dfrac{\text{tan 57°}}{\text{tan 57°}} - \dfrac{\text{cot 70°}}{\text{cot 70°}} - 1 \\[1em] \Rightarrow 2 - 1 - 1 \\[1em] \Rightarrow 0.$

Hence, $2\dfrac{\text{tan 57°}}{\text{cot 33°}} - \dfrac{\text{cot 70°}}{\text{tan 20°}} - \sqrt{2}$cos 45° = 0.