{
  "title": "Trigonometric Identities and Equations - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is sin²θ + cos²θ equal to? ✨", "options": {"A": "0", "B": "1", "C": "tanθ", "D": "cotθ"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "Simplify: 1 - sin²θ 🧮", "options": {"A": "cos²θ", "B": "tan²θ", "C": "sec²θ", "D": "cosec²θ"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "What is 1 + tan²θ equal to? 📐", "options": {"A": "cot²θ", "B": "sin²θ", "C": "sec²θ", "D": "cos²θ"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Simplify: secθ × cosθ 🔄", "options": {"A": "0", "B": "2", "C": "1", "D": "tanθ"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "What is cosecθ × sinθ? 🔢", "options": {"A": "0", "B": "cosθ", "C": "2", "D": "1"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "Simplify: tanθ × cotθ 🌟", "options": {"A": "1", "B": "0", "C": "sinθ", "D": "cosθ"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "What is 1 + cot²θ equal to? 📏", "options": {"A": "tan²θ", "B": "cosec²θ", "C": "sec²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Convert: tanθ = ? ÷ cosθ ❓", "options": {"A": "cotθ", "B": "sinθ", "C": "secθ", "D": "cosecθ"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Simplify: (sinθ/cosθ) × cosθ 🔀", "options": {"A": "tanθ", "B": "sinθ", "C": "1", "D": "cosθ"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "What is sec²θ - tan²θ? 🎯", "options": {"A": "0", "B": "2", "C": "1", "D": "sinθ"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Simplify: cosec²θ - cot²θ ⚡", "options": {"A": "tanθ", "B": "1", "C": "0", "D": "sinθ"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "Convert: cotθ = cosθ ÷ ? 🤔", "options": {"A": "tanθ", "B": "sinθ", "C": "secθ", "D": "cosecθ"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Simplify: (1 - cos²θ) ÷ sin²θ 🔄", "options": {"A": "0", "B": "1", "C": "tanθ", "D": "cotθ"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "What is sin(90° - θ) equal to? 📊", "options": {"A": "cosθ", "B": "tanθ", "C": "cotθ", "D": "sinθ"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "What is cos(90° - θ) equal to? 🎲", "options": {"A": "sinθ", "B": "cosθ", "C": "tanθ", "D": "cotθ"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Simplify: tan(90° - θ) 🧩", "options": {"A": "cotθ", "B": "tanθ", "C": "sinθ", "D": "cosθ"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "What is cot(90° - θ) equal to? 🔷", "options": {"A": "sinθ", "B": "cosθ", "C": "tanθ", "D": "cotθ"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "Simplify: sec(90° - θ) 🔶", "options": {"A": "cosecθ", "B": "secθ", "C": "cosθ", "D": "sinθ"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "What is cosec(90° - θ)? 🔺", "options": {"A": "sinθ", "B": "cosθ", "C": "secθ", "D": "tanθ"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "If sinθ = 3/5, what is cosθ? 📈", "options": {"A": "4/5", "B": "3/4", "C": "5/4", "D": "4/3"}, "correct_answer": "A"},

    {"id": 21, "difficulty": "medium", "question": "Prove: (1 - sinθ)(1 + sinθ) = cos²θ 🧠", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Solve: 2sinθ = 1 for 0° ≤ θ ≤ 90° 🔍", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "If tanθ = √3, find θ (0° ≤ θ ≤ 90°) 🧮", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "Simplify: sin⁴θ - cos⁴θ ÷ sin²θ - cos²θ 🔄", "options": {"A": "1", "B": "0", "C": "sin²θ + cos²θ", "D": "sinθ + cosθ"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Solve: √3 tanθ = 1 (0° ≤ θ ≤ 90°) ⚡", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 0°"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Prove: tanθ + cotθ = secθ × cosecθ 📐", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot prove"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "If sinθ = 0.6, find tanθ without calculator 🎯", "options": {"A": "0.75", "B": "0.8", "C": "1.25", "D": "0.6"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Solve: 2cos²θ - 1 = 0 (0° ≤ θ ≤ 90°) 🔢", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "Simplify: (sinθ + cosθ)² + (sinθ - cosθ)² 🌟", "options": {"A": "1", "B": "2", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "If 3tanθ = 1, find sinθ and cosθ values 📊", "options": {"A": "sin=1/√10, cos=3/√10", "B": "sin=3/√10, cos=1/√10", "C": "sin=1/2, cos=√3/2", "D": "sin=√3/2, cos=1/2"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Prove: (cosecθ - sinθ)(secθ - cosθ) = 1/(tanθ + cotθ) 🔄", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Solve: 4sin²θ = 3 (0° ≤ θ ≤ 90°) 🧩", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "Simplify: (1 + tanθ)² + (1 - tanθ)² 🔷", "options": {"A": "2sec²θ", "B": "2cos²θ", "C": "2sin²θ", "D": "2tan²θ"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If cotθ = 7/24, find sinθ and cosθ 🎲", "options": {"A": "sin=24/25, cos=7/25", "B": "sin=7/25, cos=24/25", "C": "sin=25/24, cos=25/7", "D": "sin=1, cos=1"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Solve: √2 sinθ = 1 (0° ≤ θ ≤ 90°) 🔺", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "Prove: (sinθ + cosecθ)² + (cosθ + secθ)² = tan²θ + cot²θ + 7 🤔", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Simplify: sinθ/(1 + cosθ) + (1 + cosθ)/sinθ 📏", "options": {"A": "2cosecθ", "B": "2secθ", "C": "2tanθ", "D": "2cotθ"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If 5sinθ = 4, find 5cosθ - 3 value 🔶", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Solve: tan²θ = 3 (0° ≤ θ ≤ 90°) 📈", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "Prove: (1 + cot²θ)(1 - cosθ)(1 + cosθ) = 1 🔍", "options": {"A": "True", "B": "False", "C": "Only for θ=30°", "D": "Cannot prove"}, "correct_answer": "A"},

    {"id": 41, "difficulty": "hard", "question": "Solve: sinθ + cosθ = √2 (0° ≤ θ ≤ 90°) 🧠", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "hard", "question": "Prove: tanA/(1 - cotA) + cotA/(1 - tanA) = 1 + secA × cosecA 🔥", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Cannot prove"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "If sinθ + cosθ = p, find sinθ × cosθ in terms of p 📐", "options": {"A": "(p² - 1)/2", "B": "(1 - p²)/2", "C": "p²/2", "D": "√(1 - p²)"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "Solve: 2sin²θ + 3cosθ = 3 (0° ≤ θ ≤ 90°) 🎯", "options": {"A": "θ = 0°", "B": "θ = 30°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "hard", "question": "Prove: (sinA - cosA + 1)/(sinA + cosA - 1) = (sinA + 1)/cosA ⚡", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "If tanθ + cotθ = 2, find tan⁷θ + cot⁷θ 🔄", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Solve: cosθ/(1 - sinθ) + cosθ/(1 + sinθ) = 4 (0° ≤ θ ≤ 90°) 🧩", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "hard", "question": "If x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ, prove x² + y² + z² = r² 🔷", "options": {"A": "True", "B": "False", "C": "Only when θ=45°", "D": "Cannot prove"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Prove: √((1 - sinθ)/(1 + sinθ)) = secθ - tanθ 🌟", "options": {"A": "True", "B": "False", "C": "Only for θ<90°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If sinθ + cosθ = √2 cosθ, prove cotθ = √2 + 1 🔶", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot prove"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "Solve: 3tan²θ - 1 = 0 (0° ≤ θ ≤ 90°) 📊", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "Prove: (1 + secθ)/secθ = sin²θ/(1 - cosθ) 🔢", "options": {"A": "True", "B": "False", "C": "Only for θ=60°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "If 7sin²θ + 3cos²θ = 4, find tanθ 🎲", "options": {"A": "1/√3", "B": "√3", "C": "1", "D": "1/2"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "Prove: (cosecθ - cotθ)² = (1 - cosθ)/(1 + cosθ) 🔺", "options": {"A": "True", "B": "False", "C": "Only for acute θ", "D": "Cannot prove"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Solve: 2cos²θ - √3 cosθ = 0 (0° ≤ θ ≤ 90°) 🤔", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "If tanθ = 1/√7, find (cosec²θ - sec²θ)/(cosec²θ + sec²θ) 📏", "options": {"A": "3/4", "B": "4/3", "C": "1/2", "D": "2/3"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Prove: (tanA + sinA)/(tanA - sinA) = (secA + 1)/(secA - 1) 🔄", "options": {"A": "True", "B": "False", "C": "Only for A=30°", "D": "Sometimes true"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Solve: sinθ = cosθ (0° ≤ θ ≤ 90°) 📈", "options": {"A": "θ = 30°", "B": "θ = 45°", "C": "θ = 60°", "D": "θ = 90°"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "hard", "question": "If sinθ + cosθ = √2 sin(90° - θ), find tanθ 🔍", "options": {"A": "√2 + 1", "B": "√2 - 1", "C": "1", "D": "√2"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "Prove: sin⁶θ + cos⁶θ = 1 - 3sin²θ cos²θ 🧮", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot prove"}, "correct_answer": "A"}
  ]
}