The expression square root of [(1 - sin A) / (1 + sin A)] is equal to which of the following trigonometric forms?

Introduction / Context:
This question tests your familiarity with standard trigonometric identities and algebraic manipulation of expressions involving sine, cosine, and tangent. The expression involves a square root of a fraction constructed from 1 minus sin A and 1 plus sin A. Such expressions often simplify to combinations of secant and tangent or cosecant and cotangent, which appear in many simplification and proving type problems in trigonometry.

Given Data / Assumptions:

• The expression given is sqrt((1 - sin A) / (1 + sin A)).
• Angle A is such that the trigonometric functions involved are defined and the denominator 1 + sin A is nonzero.
• We assume A is in a range where cos A is positive if needed for square root interpretation.
• We must choose the equivalent trigonometric form from the options.

Concept / Approach:
A standard method for simplifying expressions like (1 - sin A) / (1 + sin A) is to multiply numerator and denominator by the conjugate 1 - sin A or 1 + sin A. Here, multiplying both numerator and denominator by 1 - sin A uses the identity 1 - sin^2 A = cos^2 A in the denominator. After that, taking the square root will lead to a simple ratio involving cos A and sin A. Recognising this ratio as sec A minus tan A completes the simplification.

Step-by-Step Solution:
Step 1: Start with the inner fraction: (1 - sin A) / (1 + sin A). Step 2: Multiply numerator and denominator by (1 - sin A): ((1 - sin A)^2) / ((1 + sin A)(1 - sin A)). Step 3: Use the identity (1 + sin A)(1 - sin A) = 1 - sin^2 A = cos^2 A, so the fraction becomes (1 - sin A)^2 / cos^2 A. Step 4: The given expression is the square root of this, so sqrt((1 - sin A)^2 / cos^2 A) = (1 - sin A) / cos A, assuming cos A is positive. Step 5: Rewrite (1 - sin A) / cos A as 1 / cos A - sin A / cos A, which is sec A - tan A.

Verification / Alternative check:
You can numerically verify the identity by substituting a convenient angle such as A = 30 degrees. Compute the left side as sqrt((1 - sin 30) / (1 + sin 30)) and the right side as sec 30 minus tan 30. You will find that both sides give the same numerical value. This confirms that the algebraic simplification is consistent for at least one test value, increasing confidence that the identity holds generally in the allowed domain.

Why Other Options Are Wrong:

• Option b, cosec A - cot A, arises from a different identity involving sine and cosine in the denominator, and does not match the result of the conjugate method here.
• Option c, sec A + tan A, is the reciprocal of sec A - tan A in some contexts, but it is not equal to the simplified expression in this problem.
• Option d, cosec A + cot A, is unrelated to (1 - sin A) / (1 + sin A) and therefore does not match.
• Option e, (1 - sin A) / cos A, is an intermediate expression before rewriting in terms of secant and tangent, but the question asks for a standard trigonometric form, for which sec A - tan A is preferred.

Common Pitfalls:
A frequent mistake is to try to take the square root of numerator and denominator separately without simplifying, which often leads to incorrect or messy expressions. Another error is to forget the conjugate technique or misapply identities like 1 - sin^2 A = cos^2 A. Some students also incorrectly assume that sec A + tan A is always equal to or similar to expressions involving 1 - sin A, which is not generally true. Systematic use of algebraic manipulation and correct trigonometric identities prevents these errors.

Final Answer:
The expression simplifies to sec A - tan A.