{
  "title": "Trigonometric Identities (Trigonometry) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "sin²θ + cos²θ = ? 📐", "options": {"A": "0", "B": "1", "C": "tan²θ", "D": "sec²θ"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "1 + tan²θ = ? 🔺", "options": {"A": "sin²θ", "B": "cos²θ", "C": "sec²θ", "D": "cosec²θ"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "1 + cot²θ = ? ⚖️", "options": {"A": "sec²θ", "B": "cosec²θ", "C": "tan²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "tan θ = ? ➗", "options": {"A": "sin θ/cos θ", "B": "cos θ/sin θ", "C": "1/sin θ", "D": "1/cos θ"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "cot θ = ? 🔄", "options": {"A": "cos θ/sin θ", "B": "sin θ/cos θ", "C": "1/tan θ", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "sec θ = ? 🚀", "options": {"A": "1/cos θ", "B": "1/sin θ", "C": "cos θ/sin θ", "D": "sin θ/cos θ"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "cosec θ = ? 📏", "options": {"A": "1/sin θ", "B": "1/cos θ", "C": "sin θ/cos θ", "D": "cos θ/sin θ"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "sin(90°-θ) = ? 🔄", "options": {"A": "cos θ", "B": "sin θ", "C": "tan θ", "D": "cot θ"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "cos(90°-θ) = ? 🔄", "options": {"A": "sin θ", "B": "cos θ", "C": "tan θ", "D": "cot θ"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "tan(90°-θ) = ? 🔄", "options": {"A": "cot θ", "B": "tan θ", "C": "sin θ", "D": "cos θ"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "sec²θ - tan²θ = ? ➖", "options": {"A": "0", "B": "1", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "cosec²θ - cot²θ = ? ➖", "options": {"A": "0", "B": "1", "C": "tan²θ", "D": "sec²θ"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "sin θ cosec θ = ? ✖️", "options": {"A": "0", "B": "1", "C": "tan θ", "D": "cos θ"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "cos θ sec θ = ? ✖️", "options": {"A": "1", "B": "0", "C": "sin θ", "D": "tan θ"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "tan θ cot θ = ? ✖️", "options": {"A": "1", "B": "0", "C": "sin θ", "D": "cos θ"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "(1 - cos²θ) = ? ➖", "options": {"A": "sin²θ", "B": "tan²θ", "C": "sec²θ", "D": "cos²θ"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "(1 - sin²θ) = ? ➖", "options": {"A": "cos²θ", "B": "tan²θ", "C": "cot²θ", "D": "sec²θ"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "(sec²θ - 1) = ? ➖", "options": {"A": "tan²θ", "B": "cot²θ", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "(cosec²θ - 1) = ? ➖", "options": {"A": "cot²θ", "B": "tan²θ", "C": "sec²θ", "D": "cos²θ"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "sin²θ/(1 - cos²θ) = ? ➗", "options": {"A": "1", "B": "tan²θ", "C": "cot²θ", "D": "sec²θ"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "Simplify: (1 - sin θ)(1 + sin θ) 🔄", "options": {"A": "cos²θ", "B": "sin²θ", "C": "tan²θ", "D": "cot²θ"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Simplify: (sec θ - 1)(sec θ + 1) 🔄", "options": {"A": "tan²θ", "B": "cot²θ", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Simplify: (cosec θ + cot θ)(cosec θ - cot θ) 🔄", "options": {"A": "1", "B": "0", "C": "tan²θ", "D": "sec²θ"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Prove: tan θ + cot θ = sec θ cosec θ 📐", "options": {"A": "sin²θ+cos²θ/sin θ cos θ", "B": "1/sin θ cos θ", "C": "Both A & B", "D": "None"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "Prove: (1 + tan²A)/(1 + cot²A) = tan²A 🔄", "options": {"A": "sec²A/cosec²A", "B": "sin⁴A/cos⁴A", "C": "tan⁴A", "D": "All of these"}, "correct_answer": "D"},
    {"id": 26, "difficulty": "medium", "question": "Simplify: sin⁴θ - cos⁴θ ➖", "options": {"A": "sin²θ - cos²θ", "B": "1 - 2 cos²θ", "C": "2 sin²θ - 1", "D": "All of these"}, "correct_answer": "D"},
    {"id": 27, "difficulty": "medium", "question": "Prove: √[(1 - cos θ)/(1 + cos θ)] = cosec θ - cot θ 📏", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Prove: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A 📈", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "If x = a cos θ, y = b sin θ, then x²/a² + y²/b² = ? ➕", "options": {"A": "0", "B": "1", "C": "cos²θ+sin²θ", "D": "a²+b²"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "Prove: (1 + cot A - cosec A)(1 + tan A + sec A) = 2 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Simplify: (sin θ + cos θ)² + (sin θ - cos θ)² ➕", "options": {"A": "1", "B": "2", "C": "2 sin²θ", "D": "2 cos²θ"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Prove: (tan A + sin A)/(tan A - sin A) = (sec A + 1)/(sec A - 1) 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Prove: cos⁴A - sin⁴A = 2 cos²A - 1 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If tan θ + sin θ = m, tan θ - sin θ = n, then m² - n² = ? ➖", "options": {"A": "4√(mn)", "B": "4mn", "C": "2√(mn)", "D": "2mn"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Prove: (cosec A - sin A)(sec A - cos A) = 1/(tan A + cot A) 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Simplify: (1 + tan A + sec A)(1 + cot A - cosec A) ➕✖️", "options": {"A": "0", "B": "1", "C": "2", "D": "tan A + cot A"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "Prove: (1 - sin A + cos A)² = 2(1 - sin A)(1 + cos A) 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If sin θ + sin²θ = 1, then cos²θ + cos⁴θ = ? 🔄", "options": {"A": "0", "B": "1", "C": "sin θ", "D": "cos θ"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Prove: tan²θ - sin²θ = tan²θ sin²θ 📊", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Prove: cot²θ - cos²θ = cot²θ cos²θ 📈", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    
    {"id": 41, "difficulty": "hard", "question": "Prove: (1 + cot A + tan A)(sin A - cos A) = sin A tan A - cot A cos A 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If x = r sin A cos B, y = r sin A sin B, z = r cos A, then x² + y² + z² = ? ➕", "options": {"A": "r", "B": "r²", "C": "r³", "D": "r⁴"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "hard", "question": "Prove: (sin A + sec A)² + (cos A + cosec A)² = (1 + sec A cosec A)² 📈", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "If a cos³θ + 3a cos θ sin²θ = m, a sin³θ + 3a cos²θ sin θ = n, then (m+n)^(2/3) + (m-n)^(2/3) = ? 🔢", "options": {"A": "1", "B": "2a^(2/3)", "C": "2a", "D": "2a²"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "Prove: 2(sin⁶θ + cos⁶θ) - 3(sin⁴θ + cos⁴θ) + 1 = 0 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "If tan θ = a/b, then (a sin θ - b cos θ)/(a sin θ + b cos θ) = ? ➗", "options": {"A": "(a²-b²)/(a²+b²)", "B": "(b²-a²)/(a²+b²)", "C": "0", "D": "1"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "hard", "question": "Prove: (1 + sec A)/sec A = sin²A/(1 - cos A) 📊", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "If sin θ + cos θ = √2, then tan θ + cot θ = ? ➕", "options": {"A": "1", "B": "2", "C": "√2", "D": "2√2"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "Prove: cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If x = sec θ + tan θ, y = sec θ - tan θ, then xy = ? ✖️", "options": {"A": "0", "B": "1", "C": "2", "D": "sec²θ - tan²θ"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "hard", "question": "Prove: (tan A/(1 - cot A)) + (cot A/(1 - tan A)) = 1 + sec A cosec A 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "If (1 + sin A)(1 + sin B)(1 + sin C) = (1 - sin A)(1 - sin B)(1 - sin C), then each = ±? 🔍", "options": {"A": "cos A cos B cos C", "B": "sin A sin B sin C", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Prove: (sin θ - 2 sin³θ)/(2 cos³θ - cos θ) = tan θ 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "If cos θ + sin θ = √2 cos θ, prove tan θ = √2 - 1, then sec²θ = ? 📈", "options": {"A": "2-√2", "B": "4-2√2", "C": "2√2", "D": "3-2√2"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "hard", "question": "Prove: (1/(sec²θ - cos²θ) + 1/(cosec²θ - sin²θ)) sin²θ cos²θ = (1 - sin²θ cos²θ)/(2 + sin²θ cos²θ) 📊", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "If tan²θ = 1 - e², then sec θ + tan³θ cosec θ = ? 📐", "options": {"A": "(2-e²)^(3/2)", "B": "(2-e²)", "C": "√(2-e²)", "D": "(2-e²)√(2-e²)"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Prove: √(sec A - 1)/(sec A + 1) + √(sec A + 1)/(sec A - 1) = 2 cosec A 🔄", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "If x sin³θ + y cos³θ = sin θ cos θ and x sin θ = y cos θ, then x² + y² = ? ➕", "options": {"A": "0", "B": "1", "C": "sin²θ+cos²θ", "D": "tan²θ+cot²θ"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "hard", "question": "Prove: tan A/(1 + sec A) - tan A/(1 - sec A) = 2 cosec A 📊", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If cos θ/(1 + sin θ) + (1 + sin θ)/cos θ = 4, then θ = ? 🔍", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"}
  ]
}