### Instruction: Simplify
\[\frac{\sin x}{\cos x} + \frac{\cos x}{1 + \sin x}.\].### Response: I want to simplify this expression, so I need to find a common denominator for the two fractions and combine them.
The common denominator is \(\cos x (1 + \sin x)\), so I need to multiply the first fraction by \(1 + \sin x\) and the second fraction by \(\cos x\).
This gives me
\[\frac{\sin x (1 + \sin x)}{\cos x (1 + \sin x)} + \frac{\cos x \cdot \cos x}{\cos x (1 + \sin x)}.\]
Now I can combine the numerators and get
\[\frac{\sin x + \sin^2 x + \cos^2 x}{\cos x (1 + \sin x)}.\]
I notice that \(\sin^2 x + \cos^2 x = 1\), so I can simplify the numerator further and get
\[\frac{1 + \sin x}{\cos x (1 + \sin x)}.\]
Now I see that I can cancel out the common factor of \(1 + \sin x\) and get
\[\frac{1}{\cos x}.\]
This is the simplest form of the expression, and I can also write it as \(\sec x\) using the reciprocal identity.