Simplify: (tan²θ − sin²θ) / (2 + tan²θ + cot²θ)

Question

Simplify: (tan²θ − sin²θ) / (2 + tan²θ + cot²θ)

1. A. sec⁶θ
2. B. sin²θ
3. C. sin⁶θ (Correct answer)
4. D. cos²θ

Correct Answer

Option C — sin⁶θ

Detailed Solution & Explanation

The correct answer is sin⁶θ.

Key Points

• Simplify the numerator first:
  • tan²θ − sin²θ = sin²θ/cos²θ − sin²θ = sin²θ(1 − cos²θ)/cos²θ = sin⁴θ/cos²θ
• The denominator is a perfect square in disguise:
  • 2 + tan²θ + cot²θ = (tanθ + cotθ)², since (tanθ + cotθ)² = tan²θ + 2 + cot²θ
  • tanθ + cotθ = (sin²θ + cos²θ)/(sinθ cosθ) = 1/(sinθ cosθ)
  • so the denominator equals 1/(sin²θ cos²θ)
• Dividing: (sin⁴θ/cos²θ) × sin²θ cos²θ = sin⁶θ.

Additional Information

• The identity tanθ + cotθ = secθ · cosecθ = 1/(sinθ cosθ) is worth memorising; it appears constantly in simplification questions.
• The three Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
• Spotting that "2 + something + its reciprocal-square" is a perfect square is the key move — the same pattern gives (a + 1/a)² = a² + 2 + 1/a².
• Converting everything to sine and cosine is the reliable fallback when no identity is obvious, though it is usually the longer route.