Simplify the trigonometric expression √(1 + tan^2 A) divided by tan A and express the result using a standard trigonometric function.

Introduction / Context:
This question checks your understanding of basic trigonometric identities involving tan, sec, and cosec. It is a typical simplification problem where recognizing the identity 1 + tan^2 A = sec^2 A allows you to reduce a square root expression and then simplify a ratio.

Given Data / Assumptions:

- Expression: √(1 + tan^2 A) / tan A
- A is an acute angle so that all functions are well defined and positive where necessary
- We should express the final answer in terms of a simple trigonometric function of A

Concept / Approach:
The key identity is 1 + tan^2 A = sec^2 A. Taking the square root of sec^2 A gives sec A for angles where sec A is positive. Then we must divide this by tan A. Using the definitions tan A = sin A / cos A and sec A = 1 / cos A, we can rewrite the ratio and simplify it to a standard function.

Step-by-Step Solution:
Step 1: Start with the expression √(1 + tan^2 A) / tan A.Step 2: Use the identity 1 + tan^2 A = sec^2 A to get √(sec^2 A) / tan A, which simplifies to sec A / tan A for acute angles.Step 3: Write sec A as 1 / cos A and tan A as sin A / cos A. The expression becomes (1 / cos A) / (sin A / cos A).Step 4: Dividing by a fraction is the same as multiplying by its reciprocal. So we get (1 / cos A) * (cos A / sin A) = 1 / sin A.Step 5: Since 1 / sin A is cosec A, the simplified expression is cosec A.

Verification / Alternative check:
Pick a convenient angle, for example A = 45 degrees. Then tan 45° = 1, so √(1 + tan^2 45°) / tan 45° = √(1 + 1) / 1 = √2. On the other hand, cosec 45° = 1 / (sin 45°) = 1 / (√2 / 2) = √2. The numeric values agree, which confirms the simplification.

Why Other Options Are Wrong:
- sec A equals 1 / cos A, not 1 / sin A, so it does not match the simplified ratio.
- sin A is the reciprocal of cosec A, not equal to it.
- cos A is unrelated to the final ratio 1 / sin A.
- tan A is the original denominator and is not the result of the whole expression.

Common Pitfalls:
Students sometimes misremember the identity as 1 + tan^2 A = cosec^2 A, which is incorrect. Others forget to handle the square root correctly or mix up the definitions of tan and sec. Always recall tan A = sin A / cos A and sec A = 1 / cos A, and use algebraic fraction rules carefully when simplifying ratios of trigonometric functions.

Final Answer:
The simplified expression is cosec A.