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The given equation is:

\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} \equiv \sec A - \tan A
\]

We need to prove this identity.

### Step 1: Express $\sec A$ and $\tan A$ in terms of sine and cosine
\[
\sec A = \frac{1}{\cos A}, \quad \tan A = \frac{\sin A}{\cos A}
\]

Thus, the right-hand side (RHS) becomes:
\[
\sec A - \tan A = \frac{1}{\cos A} - \frac{\sin A}{\cos A} = \frac{1 - \sin A}{\cos A}
\]

### Step 2: Simplify the left-hand side (LHS)
The left-hand side (LHS) is:
\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}}
\]

### Step 3: Express $\frac{1 - \sin A}{1 + \sin A}$ in another form
We use the identity:
\[
\frac{1 - \sin A}{1 + \sin A} = \left(\frac{\cos A}{\cos A}\right)^2 = \frac{\cos^2 A}{(1 + \sin A)(1 - \sin A)} = \frac{\cos^2 A}{1 - \sin^2 A} = \frac{\cos^2 A}{\cos^2 A} = 1
\]
Thus, the LHS becomes:
\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \frac{\cos A}{\sqrt{\cos^2 A}} = \frac{1 - \sin A}{\cos A}
\]

### Conclusion
Both sides of the equation simplify to the same expression, proving the identity:

\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \sec A - \tan A
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. What are the identities for $\sin^2 A + \cos^2 A$?
2. How do you derive the identity $\sec A = \frac{1}{\cos A}$?
3. What is the relationship between secant and tangent functions?
4. How can trigonometric identities simplify complex expressions?
5. What are the steps to prove a trigonometric identity?
6. How do square roots affect trigonometric expressions?
7. Why is it important to express trigonometric functions in terms of sine and cosine?
8. What are the common methods for simplifying trigonometric expressions?

**Tip:** When proving trigonometric identities, converting everything to sine and cosine functions is often a useful strategy.

Learn how to prove the trigonometric identity $\sqrt{\frac{1 - \sin A}{1 + \sin A}} \equiv \sec A - \tan A$. This detailed explanation covers step-by-step methods, including converting trigonometric functions to sine and cosine, simplifying complex expressions, and using trigonometric identities effectively. Suitable for high school students and anyone studying trigonometry.

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