Evaluate the following :

$2 \times \Big(\dfrac{\text{cos}^2 20° + \text{cos}^2 70°}{\text{sin}^2 25° + \text{sin}^2 65°}\Big)$ - tan 45° + tan 13° tan 23° tan 30° tan 67° tan 77°

Since, angles are acute in the equation,

∴ cos(90° - θ) = sin θ, sin(90° - θ) = cos θ and tan(90° - θ) = cot θ.

Using above equations in

2 x $\Big(\dfrac{\text{cos}^2 20° + \text{cos}^2 70°}{\text{sin}^2 25° + \text{sin}^2 65°}\Big)$ - tan 45° + tan 13° tan 23° tan 30° tan 67° tan 77°

= 2 x $\Big(\dfrac{\text{cos}^2 20° + \text{cos}^2 (90 - 20)°}{\text{sin}^2 25° + \text{sin}^2 (90 - 25)°}\Big)$ - tan 45° + tan 13° tan 23° tan 30° tan (90 - 23)° tan (90 - 13)°

= 2 x $\Big(\dfrac{\text{cos}^2 20° + \text{sin}^2 20°}{\text{sin}^2 25° + \text{cos}^2 25°}\Big)$ - tan 45° + tan 13° tan 23° tan 30° cot 23° cot 13°

= 2 x 1 - 1 + tan 13° cot 13° tan 23° cot 23° tan 30°

= 1 + 1 x $\dfrac{1}{\sqrt{3}}$

= $\dfrac{\sqrt{3} + 1}{\sqrt{3}}$

= $\dfrac{(\sqrt{3} + 1)\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

= $\dfrac{3 + \sqrt{3}}{\sqrt{3}}$.

Hence, the value of the expression is $\dfrac{3 + \sqrt{3}}{\sqrt{3}}$.