### Instruction: Simplify
\[\frac{\sin x}{\cos x} + \frac{\cos x}{1 + \sin x}.\].### Response: This problem asks me to simplify an expression involving trigonometric functions, so I will look for ways to manipulate the fractions and use trigonometric identities.
I notice that the denominators of the two fractions are related by a Pythagorean identity: $\cos^2 x + \sin^2 x = 1$.
This suggests that I can combine the fractions by finding a common denominator.
To do this, I will multiply the first fraction by $1 + \sin x$ and the second fraction by $\cos x$, both over themselves.
This gives me
\[\frac{\sin x}{\cos x} + \frac{\cos x}{1 + \sin x} = \frac{\sin x (1 + \sin x)}{\cos x (1 + \sin x)} + \frac{\cos x (\cos x)}{\cos x (1 + \sin x)}.\]
Now I can add the numerators and simplify the expression.
I get
\[\frac{\sin x + \sin^2 x + \cos^2 x}{\cos x (1 + \sin x)} = \frac{1 + \sin x}{\cos x (1 + \sin x)}.\]
I can cancel out the factor of $1 + \sin x$ from the numerator and denominator, and I get
\[\frac{1}{\cos x}.\]
This is the simplest form of the expression, and I can also write it as $\sec x$, the reciprocal of $\cos x$.