Difficulty: Easy
Correct Answer: cos A
Explanation:
Introduction / Context:
This question tests knowledge of a fundamental trigonometric identity connecting tangent and secant, and how to simplify expressions involving radicals and trigonometric functions. Recognizing these standard identities allows very quick simplification in many exam problems.
Given Data / Assumptions:
Concept / Approach:
Recall the Pythagorean identity:
1 + tan^2 A = sec^2 A
Also, sec A is defined as 1 / cos A. Using this, we can rewrite the denominator and simplify the radical expression to obtain a simple function of A, up to sign.
Step-by-Step Solution:
Start with x = 1/√(1 + tan^2 A)
Use identity 1 + tan^2 A = sec^2 A
Then x = 1/√(sec^2 A)
The square root of sec^2 A is |sec A|, so x = 1/|sec A|
Since sec A = 1 / cos A, we have 1/|sec A| = |cos A|
In many standard exam settings, for principal values, this is taken as cos A
Therefore, x simplifies to cos A
Verification / Alternative check:
Take A = 45 degrees. Then tan 45° = 1, so 1 + tan^2 45° = 1 + 1 = 2. The left side is 1/√2. On the right side, cos 45° is also 1/√2. Thus for this specific angle, the expression equals cos A, which supports the identity based derivation.
Why Other Options Are Wrong:
Option b, sin A, is related to cosine by sin^2 A + cos^2 A = 1, but does not directly match this expression. Option c, cosec A, is 1/sin A and would make the expression larger instead of smaller. Option d, sec A, equals √(1 + tan^2 A) instead of its reciprocal. Option e, tan A, appears inside the expression but is not equal to the simplified form.
Common Pitfalls:
A common mistake is to think √(1 + tan^2 A) simplifies to 1 + tan A, which is incorrect. Another is forgetting that sec A is 1 / cos A and mixing up sec A with cos A directly. Always use the exact identity 1 + tan^2 A = sec^2 A and carefully handle the square root and reciprocal.
Final Answer:
The correct simplified value is cos A.
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