If x is defined by the trigonometric expression x = tan A − cot A, which of the following options correctly expresses x in terms of angle A using standard trigonometric identities?

Introduction / Context:
This question assesses your ability to manipulate trigonometric expressions and use basic identities to simplify them. Expressing an expression like tan A − cot A in terms of sine, cosine, and cos 2A is a common step in more advanced trigonometric simplifications used in aptitude and competitive exams.

Given Data / Assumptions:

• x is defined as x = tan A − cot A.
• Angle A is such that tan A and cot A are defined, so sin A and cos A are non zero.
• We need to re express x in a simplified form that uses sin A, cos A, and cos 2A.

Concept / Approach:
We first rewrite tan A as sin A / cos A and cot A as cos A / sin A. Then we combine these fractions over a common denominator. The numerator will involve sin^2 A and cos^2 A, which can be connected to cos 2A using the identity cos 2A = cos^2 A - sin^2 A. Finally, we arrange the terms to match one of the options given.

Step-by-Step Solution:
Write tan A as sinA / cosA and cot A as cosA / sinA.Compute x = tan A − cot A = (sinA / cosA) − (cosA / sinA).Combine over a common denominator sinA cosA: x = (sin^2A − cos^2A) / (sinA cosA).Recall that cos 2A = cos^2A − sin^2A, which implies sin^2A − cos^2A = -cos 2A.Substitute into the expression: x = -cos2A / (sinA cosA).

Verification / Alternative check:
We can check the simplification using a specific angle. Take A = 45 degrees, where sin 45° = cos 45° = 1/sqrt(2). Then tan 45° = 1 and cot 45° = 1, so x = tan A − cot A = 1 − 1 = 0. Now evaluate -cos2A / (sinA cosA). Here, 2A = 90 degrees, so cos 90° = 0. Therefore, -cos 90° = 0 and the entire fraction -cos2A / (sinA cosA) equals 0, matching the original value of x. This consistency supports our derivation.

Why Other Options Are Wrong:

• Option a, (1 + 2cos2A)/(sinA cosA), introduces an extra constant and incorrect coefficient, not supported by the derivation.
• Option b, (sinA cosA)/(1 - 2cos2A), places sinA cosA in the numerator and uses an unrelated denominator.
• Option d, (1 - 2cos2A)/(sinA cosA), again adds constants in a way that does not arise from combining tan A and cot A.
• Option e, (1 + cos2A)/(sinA cosA), incorrectly mixes a constant and cos2A instead of directly using the difference of squares sin^2A − cos^2A.

Common Pitfalls:

• Forgetting to use a common denominator when subtracting the two fractions.
• Misapplying the identity for cos 2A or confusing it with 1 − 2sin^2A or 2cos^2A − 1.
• Dropping the negative sign when replacing sin^2A − cos^2A with -cos 2A.

Final Answer:
-cos2A/(sinA cosA)