Which of the following trigonometric expressions is exactly equal to tan(A / 2) for an angle A, assuming all trigonometric functions involved are defined?

Introduction / Context: This question tests your knowledge of standard half angle formulas in trigonometry. The function tan(A / 2) can be expressed in several equivalent ways in terms of sine and cosine of A. However, in a multiple choice setting, we usually look for one canonical form. Recognising the correct half angle expression is important in many simplification and equation solving problems in trigonometry.

Given Data / Assumptions:

• We are asked to identify an expression that equals tan(A / 2).
• Angle A is such that all trigonometric functions used are defined.
• Options involve combinations of sin A and cos A.

Concept / Approach: The standard half angle identities for tangent are:

• tan(A / 2) = (1 - cos A) / sin A, and
• tan(A / 2) = sin A / (1 + cos A).

In this question, we select the exact expression that matches tan(A / 2) without simultaneously matching several options. Option A clearly matches the first canonical form directly. Other options can be analysed by simplifying them in terms of sin A and cos A to see whether they reduce to tan(A / 2) or not.

Step-by-Step Solution: Recall the half angle identity: tan(A / 2) = (1 - cos A) / sin A. Look at option A: (1 - cos A) / sin A. Option A is exactly the same as the identity, so it equals tan(A / 2). Check option B: sin A / (1 - cos A). This is the reciprocal of option A only if cos A is replaced by -cos A; it does not match the known half angle formula. Check option C: (1 + cos A) / sin A. This is closer to the reciprocal of sin A / (1 + cos A), not tan(A / 2). Check option D: 1 / (sec A + cot A). Writing in sine and cosine gives 1 / (1 / cos A + cos A / sin A), which does not simplify to tan(A / 2). Check option E: cos A / (1 + sin A). This also does not match any standard half angle form for tangent. Therefore, only option A matches the standard half angle identity for tangent.

Verification / Alternative check: We can verify using a specific angle, for example A = 60°. Then A / 2 = 30°, so tan(A / 2) = tan 30° = 1 / sqrt(3). Compute option A: (1 - cos 60°) / sin 60° = (1 - 1/2) / (sqrt(3) / 2) = (1/2) / (sqrt(3) / 2) = 1 / sqrt(3). This matches tan 30°. Evaluating other options at A = 60° gives different values, confirming that only option A equals tan(A / 2).

Why Other Options Are Wrong: Option B gives the reciprocal of option A, which corresponds to cot(A / 2) rather than tan(A / 2). Option C is not one of the canonical half angle formulas and evaluates to a different value at test angles. Option D simplifies to an expression that involves both sine and cosine in a more complex manner and does not match tan(A / 2). Option E also fails to match the half angle identity when tested with specific angles.

Common Pitfalls: A common mistake is to misremember which way the sine and cosine go in the half angle formulas, leading to confusion between tan(A / 2) and cot(A / 2). Another error is to assume that any ratio of linear combinations of sine and cosine might represent tan(A / 2) without checking the actual identity. Always refer back to the standard forms and, if unsure, test with a simple angle like 60° to distinguish between options.

Final Answer: The expression that equals tan(A / 2) is (1 - cos A) / sin A.