We can write \begin{align*} \frac{\sin x}{\cos x} + \frac{\cos x}{1 + \sin x} &= \frac{\sin x}{\cos x} + \frac{\cos x (1 - \sin x)}{(1 + \sin x)(1 - \sin x)} \\ &= \frac{\sin x}{\cos x} + \frac{\cos x (1 - \sin x)}{1 - \sin^2 x} \\ &= \frac{\sin x}{\cos x} + \frac{\cos x (1 - \sin x)}{\cos^2 x} \\ &= \frac{\sin x}{\cos x} + \frac{1 - \sin x}{\cos x} \\ &= \frac{1}{\cos x} = \boxed{\sec x}.\end{align*}