Using standard trigonometric identities, what is the simplified value of the expression sec^4 θ − sec^2 θ tan^2 θ, expressed in terms of a single basic trigonometric function of θ?

Introduction / Context:
This problem tests knowledge of basic trigonometric identities and algebraic simplification. The expression involves powers of secant and tangent, and the goal is to rewrite it as a simpler single trigonometric function. Such simplifications are common in many trigonometry questions in aptitude tests.

Given Data / Assumptions:

• The expression to simplify is sec^4 θ − sec^2 θ tan^2 θ.
• Angle θ is such that all trigonometric functions sec θ and tan θ are defined.
• We must express the result in terms of a single standard trigonometric function.

Concept / Approach:
We use the Pythagorean identity tan^2 θ + 1 = sec^2 θ. The idea is to factor the expression and then replace tan^2 θ with sec^2 θ − 1 or vice versa. This reduces the expression step by step to something much simpler. Algebraic factoring is an essential step here.

Step-by-Step Solution:
Start with the expression E = sec^4 θ − sec^2 θ tan^2 θ.Factor out sec^2 θ from both terms: E = sec^2 θ(sec^2 θ − tan^2 θ).Use the identity tan^2 θ = sec^2 θ − 1. Then sec^2 θ − tan^2 θ = sec^2 θ − (sec^2 θ − 1) = 1.Substitute this back: E = sec^2 θ * 1 = sec^2 θ.Therefore, the simplified value of the expression is sec^2 θ.

Verification / Alternative check:
Pick a convenient angle, for example θ = 45 degrees. Then sec 45° = sqrt(2), tan 45° = 1. Compute E directly: sec^4 θ = (sqrt(2))^4 = 4, and sec^2 θ tan^2 θ = (2)(1) = 2. So E = 4 − 2 = 2. Now compute sec^2 θ at 45 degrees: sec^2 45° = 2. Both results match, verifying that E simplifies correctly to sec^2 θ for at least one angle. The identity ensures this holds for all valid θ.

Why Other Options Are Wrong:

• cosec^2 θ and cot^2 θ involve reciprocal functions of sine, which do not appear naturally from sec and tan in this context.
• sec θ tan θ and tan^2 θ have different powers and cannot match sec^4 θ − sec^2 θ tan^2 θ after correct simplification.
• Only sec^2 θ emerges directly when the Pythagorean identity and factoring are applied correctly.

Common Pitfalls:

• Failing to factor out sec^2 θ at the beginning, which makes simplification more difficult.
• Misusing the identity tan^2 θ + 1 = sec^2 θ or writing it incorrectly.
• Attempting to convert everything to sine and cosine unnecessarily, which may lead to algebraic mistakes.

Final Answer:
sec^2 θ