A simplified value of $$\left(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta}\right)\left(\frac{1}{\tan \theta + \cot \theta}\right)$$ is:

$$\left(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta}\right)\left(\frac{1}{\tan \theta + \cot \theta}\right)$$

Taking LCM and replace $$\tan \theta=\frac{\sin \theta}{\cos \theta}$$, $$\cot \theta=\frac{\cos \theta}{\sin \theta}$$

$$\left(\frac{\sin^2 \theta+(1 + \cos \theta)^2}{\sin\theta(1 + \cos \theta)} \right)\left(\frac{1}{\frac{\sin\theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}\right)$$

$$\left(\frac{\sin^2 \theta+(1 + \cos \theta)^2}{\sin\theta(1 + \cos \theta)} \right)\left(\frac{1}{\frac{\sin^2\theta + \cos^2 \theta}{\cos\theta\sin \theta}}\right)$$
$$\left(\frac{\sin^2 \theta+(1 + \cos \theta)^2}{\sin\theta(1 + \cos \theta)} \right)\left(\frac{\cos\theta\sin \theta}{\sin^2\theta + \cos^2 \theta}\right)$$

$$\left(\frac{\sin^2 \theta+(1 + \cos \theta)^2}{\sin\theta(1 + \cos \theta)} \right)\left({\cos\theta\sin \theta}\right)$$

$$\left(\frac{\sin^2 \theta+(1 + \cos^2 \theta+2\cos\theta)}{(1+\cos\theta)} \right)\left({\cos\theta}\right)$$

$$\left(\frac{2+2\cos\theta}{(1+\cos\theta)} \right)\left({\cos\theta}\right)$$

$$2\cos\theta$$