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If (1 + sec A) / tan A = x, then x is equal to which trigonometric expression?

Difficulty: Medium

Correct Answer: cot(A/2)

Explanation:


Introduction / Context:
This question focuses on trigonometric simplification and half angle identities. The expression (1 + sec A) / tan A can look complicated, but by expressing everything in terms of sine and cosine and then using standard double angle and half angle relationships, we can convert it into a simple form involving A/2. The goal is to match this simplified result with one of the given options.


Given Data / Assumptions:

  • (1 + sec A) / tan A = x.
  • sec A = 1 / cos A.
  • tan A = sin A / cos A.
  • A is such that all trigonometric functions involved are defined.


Concept / Approach:
The main idea is to convert sec A and tan A into sine and cosine, simplify the fraction, and then recognize a standard half angle identity. The combination (1 + cos A) / sin A is known to simplify to cot(A/2), which is the key identity used here.


Step-by-Step Solution:

Step 1: Rewrite sec A as 1 / cos A and tan A as sin A / cos A. Step 2: Substitute into the expression: x = (1 + 1/cos A) / (sin A / cos A). Step 3: Combine the numerator: 1 + 1/cos A = (cos A + 1) / cos A. Step 4: Now divide by sin A / cos A, which is the same as multiplying by cos A / sin A. Step 5: So x = [(cos A + 1) / cos A] * [cos A / sin A] = (cos A + 1) / sin A. Step 6: Use half angle identity: 1 + cos A = 2*cos^2(A/2) and sin A = 2*sin(A/2)*cos(A/2). Step 7: Substitute: (cos A + 1) / sin A = [2*cos^2(A/2)] / [2*sin(A/2)*cos(A/2)]. Step 8: Cancel common factors 2 and cos(A/2): x = cos(A/2) / sin(A/2) = cot(A/2).


Verification / Alternative check:
You can verify the identity numerically by choosing a specific angle such as A = 60 degrees. Compute both sides using a calculator with enough precision. The numeric value of (1 + sec 60) / tan 60 will match the value of cot 30, confirming the derivation.


Why Other Options Are Wrong:

tan(A/2): is the reciprocal of the correct result. cosec(A/2) and sec(A/2): involve reciprocals of sine or cosine of half angle and do not match the simplified ratio. cot A: corresponds to cos A / sin A, not the half angle expression derived here.


Common Pitfalls:
A common mistake is to forget to convert both sec A and tan A into sine and cosine consistently, or to overlook the half angle forms of 1 ± cos A and sin A. Another pitfall is trying to manipulate directly in terms of sec and tan without reducing to basic sin and cos, which makes the simplification much harder.


Final Answer:
cot(A/2)

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