If $$\cot \theta + \tan\theta = 2 \sec \theta, 0^\circ < \theta < 90^\circ $$, then the value of $$\frac{\tan 2 \theta - \sec \theta}{\cot 2 \theta + \cosec \theta}$$

$$\cot \theta + \tan\theta = 2 \sec \theta$$

On putting the value of $$\theta$$ = 30$$\degree$$

$$\cot30\degree + \tan30\degree = 2 \sec30\degree$$

$$ \sqrt3 + \frac{1}{\sqrt3} = 2 \times \frac{2}{\sqrt3}$$

$$\frac{4}{\sqrt3} = \frac{4}{\sqrt3}$$

$$\frac{\tan 2 \theta - \sec \theta}{\cot 2 \theta + \cosec \theta}$$

= $$\frac{\tan 2 \times 30\degree - \sec30\degree }{\cot 2 \times 30\degree + \cosec30\degree }$$

= $$\frac{\sqrt{\ 3}-\frac{2}{\sqrt{\ 3}}}{\frac{1}{\sqrt{\ 3}}+2}$$

= $$\frac{\frac{1}{\sqrt{\ 3}}}{\frac{1 + 2\sqrt3}{\sqrt{\ 3}}}$$

= $$\frac{1}{1 + 2\sqrt3} \times \frac{1 - 2\sqrt3}{1 - 2\sqrt3}$$

= $$\frac{1}{1 + 2\sqrt3} \times \frac{1 - 2\sqrt3}{1^2 - (2\sqrt3)^2}$$

= $$\frac{2\sqrt3 - 1}{11}$$