If tan θ + cot θ = x (with θ such that tan θ and cot θ are defined), express tan^4 θ + cot^4 θ purely in terms of x.

Introduction / Context: This question tests algebraic identities with tan θ and cot θ treated as reciprocal variables. When you know tan θ + cot θ, you can derive tan^2 θ + cot^2 θ by squaring, and then use another squaring step to reach tan^4 θ + cot^4 θ.

Given Data / Assumptions:

• tan θ + cot θ = x • tan θ and cot θ are defined (so sin θ and cos θ are not zero) • Required: tan^4 θ + cot^4 θ in terms of x only

Concept / Approach: Let t = tan θ, so cot θ = 1/t. Then: x = t + 1/t. Square to get t^2 + 1/t^2, then square again to get t^4 + 1/t^4. Use: (t + 1/t)^2 = t^2 + 2 + 1/t^2 and (t^2 + 1/t^2)^2 = t^4 + 2 + 1/t^4.

Step-by-Step Solution: 1) Let t = tan θ, so cot θ = 1/t 2) Given: x = t + 1/t 3) Square both sides: x^2 = (t + 1/t)^2 = t^2 + 2 + 1/t^2 4) So: t^2 + 1/t^2 = x^2 − 2 5) Square again: (t^2 + 1/t^2)^2 = (x^2 − 2)^2 6) Expand left side: (t^2 + 1/t^2)^2 = t^4 + 2 + 1/t^4 7) Therefore: t^4 + 1/t^4 = (x^2 − 2)^2 − 2 8) Expand: (x^2 − 2)^2 − 2 = (x^4 − 4x^2 + 4) − 2 = x^4 − 4x^2 + 2 9) Rewrite as: x^2(x^2 − 4) + 2

Verification / Alternative check: If θ = 45°, tan θ = 1 and cot θ = 1, so x = 2. Then tan^4 + cot^4 = 1 + 1 = 2. Formula gives x^2(x^2 − 4) + 2 = 4*(0) + 2 = 2. Correct.

Why Other Options Are Wrong: • x^2 − 2 corresponds to tan^2 + cot^2, not fourth powers. • x or x^4 − 2x + 4 do not match the correct identity expansion.

Common Pitfalls: • Forgetting the +2 term when squaring (t + 1/t)^2. • Forgetting to subtract 2 after squaring (t^2 + 1/t^2)^2.

Final Answer: x^2(x^2 − 4) + 2