The expression you provided is:

$\frac{1 + \cos{\alpha}}{1 + \tan{\alpha}}$

To simplify this expression, let's express $\tan{\alpha}$ in terms of $\sin{\alpha}$ and $\cos{\alpha}$:

$\tan{\alpha} = \frac{\sin{\alpha}}{\cos{\alpha}}$

Substitute this into the expression:

$\frac{1 + \cos{\alpha}}{1 + \frac{\sin{\alpha}}{\cos{\alpha}}}$

Now simplify the denominator by getting a common denominator:

$1 + \frac{\sin{\alpha}}{\cos{\alpha}} = \frac{\cos{\alpha} + \sin{\alpha}}{\cos{\alpha}}$

Substituting this back into the expression:

$\frac{1 + \cos{\alpha}}{\frac{\cos{\alpha} + \sin{\alpha}}{\cos{\alpha}}}$

Now, multiply the numerator by the reciprocal of the denominator:

$\frac{1 + \cos{\alpha}}{1 + \frac{\sin{\alpha}}{\cos{\alpha}}} = \frac{1 + \cos{\alpha}}{1} \times \frac{\cos{\alpha}}{\cos{\alpha} + \sin{\alpha}}$

Simplifying:

$\frac{(1 + \cos{\alpha}) \cdot \cos{\alpha}}{\cos{\alpha} + \sin{\alpha}}$

So the simplified form of the expression is:

$\frac{(1 + \cos{\alpha}) \cdot \cos{\alpha}}{\cos{\alpha} + \sin{\alpha}}$

Would you like a deeper explanation or have any other questions?

Here are five related questions:

1. How can we simplify trigonometric expressions involving sine and cosine?
2. What is the relationship between $\tan{\alpha}$, $\sin{\alpha}$, and $\cos{\alpha}$?
3. How do you factor trigonometric expressions?
4. Can the identity $\sin^2 \alpha + \cos^2 \alpha = 1$ help in simplifying other expressions?
5. What happens when $\alpha = 0$ or $\alpha = \pi/2$ in this expression?

Tip: Remember, converting between trigonometric functions using identities like $\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$ can greatly simplify complex expressions!