In trigonometry, what is the simplified value of √[(1 + sin A)/(1 - sin A)]?

Introduction / Context:
This question is a standard trigonometric simplification involving radicals and ratios of sine and cosine. Expressions like √[(1 + sin A)/(1 - sin A)] frequently appear in exam problems, and there is a well known identity that simplifies this expression to a combination of sec A and tan A.

Given Data / Assumptions:

• Expression: √[(1 + sin A)/(1 - sin A)]
• A is an angle such that the expression is defined.
• We must express the result in terms of standard functions sec A and tan A.

Concept / Approach:
We can rationalize the expression by multiplying numerator and denominator inside the radical by (1 + sin A). We also use the identity 1 - sin^2 A = cos^2 A. After simplification, the radical becomes a square of a rational expression in sine and cosine, which can be converted into sec A and tan A.

Step-by-Step Solution:
Start with √[(1 + sin A)/(1 - sin A)] Multiply numerator and denominator inside the root by (1 + sin A): (1 + sin A)/(1 - sin A) = [(1 + sin A)^2] / [(1 - sin A)(1 + sin A)] Denominator becomes 1 - sin^2 A = cos^2 A So the expression inside the root is [(1 + sin A)^2] / [cos^2 A] Take the square root: √[(1 + sin A)^2 / cos^2 A] = (1 + sin A) / cos A Now write (1 + sin A) / cos A as (1 / cos A) + (sin A / cos A) = sec A + tan A

Verification / Alternative check:
Take a specific angle, for example A = 30 degrees. The left side is √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3. The right side is sec 30° + tan 30° = (2/√3) + (1/√3) = 3/√3 = √3. Both sides match, confirming the identity.

Why Other Options Are Wrong:
Option a, sec A - tan A, is related to the conjugate but not equal to this expression. Option b introduces cosec A and tan A, which do not arise from the algebraic steps. Option d, cosec A - tan A, mixes functions in a way that does not match the derived form. Option e, sec A - cot A, also does not follow from the simplification process.

Common Pitfalls:
Students sometimes expand (1 + sin A)^2 incorrectly or forget the identity 1 - sin^2 A = cos^2 A. Another frequent error is taking the square root of a fraction but not applying it to both numerator and denominator correctly. Carefully performing each algebraic step and using identities in the right order prevents these mistakes.

Final Answer:
The simplified value of the expression is sec A + tan A.