To simplify the given expression:

$\frac{\cos x}{1 - \sin x} + \frac{1 - \sin x}{\cos x}$

We proceed as follows:

Step 1: Combine the fractions

The least common denominator (LCD) of the two fractions is $(1 - \sin x)(\cos x)$. Rewrite the terms with the common denominator:

$\frac{\cos x}{1 - \sin x} + \frac{1 - \sin x}{\cos x} = \frac{\cos^2 x}{(1 - \sin x)(\cos x)} + \frac{(1 - \sin x)^2}{(1 - \sin x)(\cos x)}.$

Combine the numerators:

$\frac{\cos^2 x + (1 - \sin x)^2}{(1 - \sin x)(\cos x)}.$

Step 2: Expand $(1 - \sin x)^2$

$(1 - \sin x)^2 = 1 - 2\sin x + \sin^2 x.$

Substitute this back into the numerator:

$\cos^2 x + (1 - 2\sin x + \sin^2 x).$

Step 3: Use the Pythagorean Identity

From the identity $\cos^2 x + \sin^2 x = 1$, replace $\cos^2 x + \sin^2 x$ with $1$:

$1 + 1 - 2\sin x = 2 - 2\sin x.$

So the numerator becomes $2 - 2\sin x$, and the expression is now:

$\frac{2 - 2\sin x}{(1 - \sin x)(\cos x)}.$

Step 4: Simplify the numerator

Factor out $2$ from the numerator:

$\frac{2(1 - \sin x)}{(1 - \sin x)(\cos x)}.$

Cancel $(1 - \sin x)$ in the numerator and denominator (valid as long as $1 - \sin x \neq 0$):

$\frac{2}{\cos x}.$

Step 5: Final Simplification

$\frac{2}{\cos x} = 2 \sec x.$

Final Answer:

The simplified expression is:

D) $2 \sec x$.

Would you like a further breakdown or have any related questions? Here are five related questions you can explore:

1. What happens if $1 - \sin x = 0$ (undefined cases in the original expression)?
2. How does this simplification apply to solving trigonometric equations?
3. What are some common errors to avoid when simplifying trigonometric fractions?
4. Can you derive an alternate form of the original expression using tangent or cosecant?
5. How can we graphically interpret $\sec x$ and its transformations?

Tip: Always simplify trigonometric expressions step by step using identities like Pythagorean, reciprocal, and co-function relationships.