If secA + tanA = x, then which expression in terms of sinA correctly represents x?

Introduction / Context:
This problem tests the ability to convert an expression involving secA and tanA into a form involving only sinA. Such manipulations are common in trigonometric simplification and are often used in competitive examinations to check understanding of identities and algebraic transformations of trigonometric ratios.

Given Data / Assumptions:

• secA + tanA = x.
• We must express x in terms of sinA using a single square root expression.
• A is taken so that all required trigonometric functions are defined.

Concept / Approach:
We start by expressing secA and tanA in terms of sine and cosine. Then we factor the expression and, by squaring x, derive a relation for x^2 in terms of sinA. After simplification, we identify a standard form and finally take the positive square root to obtain x. A useful identity here is that (secA + tanA)^2 can be written in terms of sinA using the relation between sin^2A and cos^2A.

Step-by-Step Solution:
Step 1: Rewrite secA and tanA using sine and cosine: secA = 1/cosA, tanA = sinA/cosA. Step 2: Add them: secA + tanA = (1 + sinA)/cosA. Step 3: Square both sides to find x^2: x^2 = [(1 + sinA)/cosA]^2. Step 4: Expand the square: x^2 = (1 + sinA)^2 / cos^2A. Step 5: Replace cos^2A with 1 - sin^2A to get x^2 = (1 + sinA)^2 / (1 - sin^2A). Step 6: Factor the denominator: 1 - sin^2A = (1 - sinA)(1 + sinA). Step 7: Cancel one factor of (1 + sinA) to obtain x^2 = (1 + sinA)/(1 - sinA). Step 8: Taking the positive square root for suitable angles, we have x = √[(1 + sinA)/(1 - sinA)].

Verification / Alternative check:
Take A = 30 degrees. On one hand, sec30 degrees = 1/cos30 degrees = 2/√3 and tan30 degrees = 1/√3, so secA + tanA = 3/√3 = √3. On the other hand, sin30 degrees = 1/2, so √[(1 + sinA)/(1 - sinA)] becomes √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3. Both expressions match, confirming the identity for this angle.

Why Other Options Are Wrong:

• Option A: √[(1 - sinA)/(1 + sinA)] gives 1/x rather than x.
• Option B: (1 + sinA)/(1 - sinA) represents x^2, not x itself.
• Option C: (1 - sinA)/(1 + sinA) is the reciprocal of Option B.
• Option E: (1 - sinA)/cosA is related to secA - tanA, not secA + tanA.

Common Pitfalls:
Learners sometimes stop after finding x^2 and forget to take the square root, leading them to choose option B instead of the correct radical form. Others may mix up the plus and minus signs in 1 ± sinA when factoring the denominator. Careful algebra and attention to each step prevent these issues.

Final Answer:
The correct expression for x is √[(1 + sinA)/(1 - sinA)].