To solve the problem given in the image, we simplify the expression:

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

Step 1: Find a common denominator

The common denominator between the two fractions is $(\sin x)(1 - \cos x)$. Rewrite each fraction with this common denominator:

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

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

Now, the expression becomes:

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

Step 2: Simplify the numerator

Expand $(1 - \cos x)^2$:

$(1 - \cos x)^2 = 1 - 2\cos x + \cos^2 x$

So the numerator is:

$1 - 2\cos x + \cos^2 x + \sin^2 x$

Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, substitute $1$ for $\sin^2 x + \cos^2 x$:

$1 - 2\cos x + 1 = 2 - 2\cos x$

Thus, the expression becomes:

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

Step 3: Factorize the numerator

Factor $2$ from the numerator:

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

Cancel $1 - \cos x$ from the numerator and denominator (provided $1 - \cos x \neq 0$):

$\frac{2}{\sin x}$

Step 4: Simplify using reciprocal identities

The reciprocal of $\sin x$ is $\csc x$. Thus, the result is:

$2 \csc x$

Final Answer:

The correct option is A. $2 \csc x$.

Would you like further clarifications or detailed steps? Here are some related questions:

1. What are the Pythagorean identities in trigonometry?
2. How does factoring simplify trigonometric expressions?
3. When is it valid to cancel terms in a fraction?
4. What are the reciprocal trigonometric functions?
5. Can this method be applied to different trigonometric equations?

Tip: Always look for opportunities to use trigonometric identities to simplify expressions efficiently.