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The problem in the image asks to prove the identity:

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

### Proof of the Identity:

1. Start with the left-hand side (LHS):

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

2. Expand \((1 + \sin x)^2\):

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

Thus, the LHS becomes:

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

3. Recognize that \(\cos^2 x = 1 - \sin^2 x\) using the Pythagorean identity:

\[
\cos^2 x = 1 - \sin^2 x
\]

4. Substitute into the LHS:

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

5. Notice that the numerator \(1 + 2 \sin x + \sin^2 x\) is a perfect square, which can be factored as:

\[
(1 + \sin x)^2
\]

So the LHS becomes:

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

6. Now use the difference of squares in the denominator:

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

7. Simplify the expression:

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

Thus, the left-hand side is equal to the right-hand side (RHS), proving the identity.

---

Would you like further clarification or details on any of the steps?

Here are 5 related questions to expand your understanding:

1. What other trigonometric identities could be derived using the Pythagorean identity?
2. Can this identity be used to simplify trigonometric integrals?
3. How would this proof change if we used different trigonometric substitutions?
4. What is the geometric interpretation of this identity?
5. How does the structure of this identity help in solving trigonometric equations?

**Tip:** Factoring expressions and using known identities like the Pythagorean identity is often the key to simplifying complex trigonometric proofs.

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

A step-by-step proof of the trigonometric identity \(\frac{(1 + \sin x)^2}{\cos^2 x} = \frac{1 + \sin x}{1 - \sin x}\), involving expansion, factoring, and use of the Pythagorean identity. Suitable for students in Grades 10-12.

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