To prove the trigonometric identity:

$\frac{1}{1 - \sin x} + \frac{1 - \sin x}{\cos^2 x} = 2 \sec^2 x$

we can proceed with the following steps, as depicted in the image:

Step 1: Rewrite the Second Fraction

Rewrite the expression $\frac{1 - \sin x}{\cos^2 x}$ by separating terms to match the standard trigonometric identities.

Step 2: Combine the Fractions Using a Common Denominator

Find a common denominator between $\frac{1}{1 - \sin x}$ and $\frac{1 - \sin x}{\cos^2 x}$. This common denominator will involve both $(1 - \sin x)$ and $\cos^2 x$.

Step 3: Apply Pythagorean Identity

Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to simplify expressions involving $\cos^2 x$ in terms of $\sin x$.

Step 4: Simplify to Show Equivalence to $2 \sec^2 x$

After combining and simplifying, aim to rewrite the expression in terms of $2 \sec^2 x$.

The image shows the steps following a structured approach, with specific algebraic and trigonometric manipulations to simplify the expression systematically.

Would you like a more detailed, step-by-step derivation of each part?

Here are 5 related questions for further exploration:

1. How can we derive similar identities using other trigonometric functions?
2. What are some common trigonometric identities that are helpful for proofs like this?
3. How does this identity relate to calculus, specifically in integration?
4. What other Pythagorean identities can be used in such trigonometric proofs?
5. How can we verify identities graphically to understand their validity?

Tip: When working with trigonometric identities, always look for opportunities to use Pythagorean identities and express everything in terms of sine and cosine for easier simplification.