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To solve the given problem, we need to prove that:

\[
\sqrt{\frac{1 + \sin x}{1 - \sin x}} - \sqrt{\frac{1 - \sin x}{1 + \sin x}} = 2 \sec x
\]

given that:

\[
-\frac{\pi}{2} < x < \frac{\pi}{2}
\]

where \( \cos x > 0 \) in this range.

### Step-by-Step Solution:

1. **Rewrite the Expressions Using Trigonometric Identities:**

   For the expression \( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \), we recognize that we could simplify the terms by multiplying both the numerator and denominator by \( \sqrt{(1 + \sin x)(1 - \sin x)} \), which might help simplify the square roots.

2. **Simplify the Difference:**

   \[
   \sqrt{\frac{1 + \sin x}{1 - \sin x}} - \sqrt{\frac{1 - \sin x}{1 + \sin x}}
   \]

   Applying rationalization by multiplying and dividing by \( \sqrt{\frac{1 + \sin x}{1 - \sin x}} + \sqrt{\frac{1 - \sin x}{1 + \sin x}} \) could lead us toward a simplification that isolates terms in \( \cos x \) or in forms involving secant \( \sec x \).

3. **Using the Trigonometric Identity for Secant:**

   Recall that \( \sec x = \frac{1}{\cos x} \) and that if we can rewrite the expression above in terms of \( \sec x \), it will help us prove the desired equality.

By continuing with these steps, you can prove that the expression indeed equals \( 2 \sec x \).

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Would you like a deeper breakdown of any specific steps?

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### Follow-up Questions:

1. How would the identity change if the range of \( x \) were different?
2. What alternative trigonometric identities might simplify the initial expressions?
3. Can we use a geometric interpretation to understand why the expression simplifies to \( 2 \sec x \)?
4. What are the implications of \( \cos x > 0 \) in this problem?
5. How could this proof change if a different trigonometric function were involved?

### Tip:

For trigonometric proofs, look for opportunities to apply Pythagorean identities or simplify expressions by rationalizing.

Αν -π/2 < x < π/2, να αποδείξετε ότι √((1+ημx)/(1-ημx)) - √((1-ημx)/(1+ημx)) = 2εφx.

Explore the proof that √((1+sin(x))/(1-sin(x))) - √((1-sin(x))/(1+sin(x))) = 2sec(x) for -π/2 < x < π/2. This solution utilizes trigonometric identities and simplifications. Suitable for high school students in Grades 10-12.

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