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To prove the identity 

\[
\frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x} = 2 \sec^2 x,
\]

we can proceed by simplifying the left side step-by-step. Here’s the solution process:

### Step 1: Simplify the Left Side
The left side of the identity is:

\[
\frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x}.
\]

We need a common denominator to combine these two fractions. The common denominator is \((1 - \sin x) \cos^2 x\).

### Step 2: Rewrite Each Fraction with the Common Denominator
Rewrite each term over the common denominator:

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

### Step 3: Combine the Fractions
Now, combine the fractions:

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

### Step 4: Expand \( (1 - \sin x)^2 \)
Expanding \( (1 - \sin x)^2 \):

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

### Step 5: Substitute and Simplify the Numerator
Substitute back into the fraction:

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

### Step 6: Use the Pythagorean Identity
Recall the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Replace \(\sin^2 x + \cos^2 x\) with \(1\):

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

### Step 7: Factor Out a 2
Factor out a 2 in the numerator:

\[
\frac{2(1 - \sin x)}{(1 - \sin x) \cos^2 x}.
\]

### Step 8: Cancel \(1 - \sin x\) from the Numerator and Denominator
Now, cancel \(1 - \sin x\) from the numerator and denominator:

\[
\frac{2}{\cos^2 x}.
\]

### Step 9: Rewrite in Terms of \(\sec^2 x\)
Since \(\frac{1}{\cos^2 x} = \sec^2 x\), we have:

\[
\frac{2}{\cos^2 x} = 2 \sec^2 x.
\]

### Conclusion
We have shown that:

\[
\frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x} = 2 \sec^2 x.
\]

Thus, the identity is proven.

Would you like a detailed explanation of each step, or do you have any specific questions?

Here are five related questions to deepen your understanding:
1. How does the Pythagorean identity help in simplifying trigonometric expressions?
2. Why is it useful to find a common denominator when combining fractions?
3. How do we determine which terms can be canceled in a fraction?
4. What are other ways to prove trigonometric identities?
5. How does understanding secant and cosine relationships assist in proving identities?

**Tip:** Always look for opportunities to apply fundamental identities, such as the Pythagorean identity, when simplifying trigonometric expressions.

Prove the identity \( \frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x} = 2 \sec^2 x \).

Step-by-step solution to prove the trigonometric identity \( \frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x} = 2 \sec^2 x \). This problem involves simplification using the Pythagorean identity and other trigonometric principles. Suitable for high school students in Grades 11-12.

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