Simplify: $$\frac{\cot \theta}{\cosec \theta + 1} + \frac{\cosec \theta + 1}{\cot \theta}$$

Correct option is C

Given:
​$$\frac{\cot \theta}{\cosec \theta + 1} + \frac{\cosec \theta + 1}{\cot \theta}$$ 
Formula Used:
​$$1+ \cot^2\theta = cosec^2 \theta$$​​
Solution:
​$$\frac{\cot \theta}{\cosec \theta + 1} + \frac{\cosec \theta + 1}{\cot \theta}\\\ \\= \frac{\cot^2 \theta + (\cosec \theta + 1)^2}{(\cosec \theta + 1) \cdot \cot \theta}\\\ \\= \frac{\cot^2 \theta + \cosec^2 \theta + 1 + 2\cosec \theta}{(\cosec \theta + 1) \cdot \cot \theta}\\\ \\= \frac{\cosec^2 \theta + \cot^2 \theta + 2\cosec \theta + 1}{(\cosec \theta + 1) \cdot \cot \theta }\\\ \\ = \frac{2\cosec \theta (\cosec \theta + 1)}{\cot \theta (\cosec \theta + 1) } \\\ \\= \frac{2\cosec \theta}{\cot \theta}\\\ \\= 2 \times \frac{1}{\sin\theta} \times \frac{\sin \theta}{\cos\theta}\\= 2 \sec \theta$$​​