If sec^2 θ + tan^2 θ = 3, find the exact value of: sec^4 θ − tan^4 θ Use standard identities and algebraic factorization to simplify the expression.

Introduction / Context:
This question tests advanced simplification using trigonometric identities and algebraic factorization. The given condition involves sec^2 θ and tan^2 θ, which are connected by the fundamental identity: sec^2 θ = 1 + tan^2 θ. Rather than trying to find θ, you can treat tan^2 θ as a variable and solve for it using the given equation. Then compute sec^2 θ from the identity. The target expression sec^4 θ − tan^4 θ is a difference of fourth powers, which can be simplified efficiently using: A^4 − B^4 = (A^2 − B^2)(A^2 + B^2). Here, A^2 corresponds to sec^2 θ and B^2 corresponds to tan^2 θ. This approach avoids any angle calculation and stays purely algebraic. It is a common “hard” aptitude pattern because it requires combining identity knowledge with the correct factorization to get a simple exact result.

Given Data / Assumptions:

• sec^2 θ + tan^2 θ = 3 • Identity: sec^2 θ = 1 + tan^2 θ • Factorization: A^4 − B^4 = (A^2 − B^2)(A^2 + B^2) • Required: sec^4 θ − tan^4 θ

Concept / Approach:
Let t = tan^2 θ. Then sec^2 θ = 1 + t. Substitute into the given: (1 + t) + t = 3 ⇒ 1 + 2t = 3 ⇒ t = 1. So tan^2 θ = 1 and sec^2 θ = 2. Now simplify: sec^4 θ − tan^4 θ = (sec^2 θ − tan^2 θ)(sec^2 θ + tan^2 θ). Compute both factors using the known values.

Step-by-Step Solution:
1) Let t = tan^2 θ. 2) Then sec^2 θ = 1 + t. 3) Substitute into the given equation: sec^2 θ + tan^2 θ = 3 (1 + t) + t = 3 4) Solve for t: 1 + 2t = 3 ⇒ 2t = 2 ⇒ t = 1 5) So: tan^2 θ = 1 and sec^2 θ = 1 + 1 = 2 6) Use factorization: sec^4 θ − tan^4 θ = (sec^2 θ − tan^2 θ)(sec^2 θ + tan^2 θ) 7) Compute each factor: sec^2 θ − tan^2 θ = 2 − 1 = 1 sec^2 θ + tan^2 θ = 3 (given) 8) Multiply: 1*3 = 3

Verification / Alternative check:
From tan^2 θ = 1, tan θ = 1 for an acute angle implies θ = 45°. Then sec^2 θ = 1 + tan^2 θ = 2, consistent with the given sum 2 + 1 = 3. Now sec^4 θ − tan^4 θ = 2^2 − 1^2 = 4 − 1 = 3, matching the factorization result exactly.

Why Other Options Are Wrong:
• 1: would occur if you mistakenly used sec^2 θ − tan^2 θ alone and ignored the sum factor. • √3 or 1/√3: unnecessary surds; the algebra forces an integer here. • 0: would require sec^2 θ = tan^2 θ, which would contradict sec^2 θ = 1 + tan^2 θ.

Common Pitfalls:
• Forgetting sec^2 θ = 1 + tan^2 θ. • Simplifying sec^4 − tan^4 as (sec^2 − tan^2)^2 (wrong). • Not using the given value of sec^2 θ + tan^2 θ = 3 when factorizing.

Final Answer:
3