{
  "title": "Simple Problems (Trigonometry) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Right triangle 📐 has sides 3,4,5. sin θ for smallest angle?", "options": {"A": "3/5", "B": "4/5", "C": "3/4", "D": "4/3"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Triangle with sides 5,12,13. cos of largest angle?", "options": {"A": "5/13", "B": "12/13", "C": "5/12", "D": "12/5"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "tan θ = 3/4, find sin θ 📏", "options": {"A": "3/5", "B": "4/5", "C": "3/4", "D": "5/3"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "sin θ = 5/13, find cos θ 🔍", "options": {"A": "12/13", "B": "5/12", "C": "13/12", "D": "12/5"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "cos θ = 0.6, find tan θ 🧮", "options": {"A": "0.8", "B": "4/3", "C": "3/4", "D": "1.25"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Triangle ABC right at B: AB=6, BC=8. sin C = ? 📐", "options": {"A": "6/10", "B": "8/10", "C": "6/8", "D": "8/6"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "If sec θ = 13/12, find sin θ 🔎", "options": {"A": "5/13", "B": "12/13", "C": "5/12", "D": "13/5"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "cosec θ = 25/24, find tan θ 📊", "options": {"A": "24/7", "B": "7/24", "C": "24/25", "D": "7/25"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "cot θ = 1, find sin θ + cos θ ➕", "options": {"A": "√2", "B": "1", "C": "2", "D": "1/√2"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "sin θ = cos θ, find θ (0°<θ<90°) 🔄", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "2 sin θ = 1, find θ 🎯", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "√3 tan θ = 1, find θ 📉", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "2 cos θ = √3, find θ 📈", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "tan θ = √3, find θ 🌟", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "sin(90°-θ) = 0.5, find θ 🔄", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "cos(90°-θ) = √3/2, find θ 🔍", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "If sin θ = 3/5, find cos θ 📏", "options": {"A": "4/5", "B": "3/4", "C": "5/4", "D": "4/3"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If cos θ = 24/25, find sin θ 🔎", "options": {"A": "7/25", "B": "24/7", "C": "7/24", "D": "25/7"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "tan θ = 5/12, find sin θ + cos θ ➕", "options": {"A": "17/13", "B": "13/17", "C": "17/15", "D": "15/17"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "sin θ = 0.8, find sec θ 🚀", "options": {"A": "5/3", "B": "3/5", "C": "4/5", "D": "5/4"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "sin θ = 12/13, find 2 sin θ cos θ ✖️", "options": {"A": "120/169", "B": "119/169", "C": "60/169", "D": "24/13"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "tan θ + cot θ = 2, find sin θ 🔄", "options": {"A": "1/√2", "B": "√3/2", "C": "1/2", "D": "1"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "sin θ - cos θ = 0, find sin²θ + cos²θ ➕", "options": {"A": "0", "B": "0.5", "C": "1", "D": "2"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "3 tan θ = √3, find sin²θ - cos²θ ➖", "options": {"A": "-0.5", "B": "0", "C": "0.5", "D": "1"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "sin θ = 1/√5, find tan θ + cot θ ➕", "options": {"A": "2.5", "B": "3", "C": "4", "D": "5"}, "correct_answer": "D"},
    {"id": 26, "difficulty": "medium", "question": "sec θ + tan θ = 3, find sin θ 🔍", "options": {"A": "0.6", "B": "0.8", "C": "0.4", "D": "0.5"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "cosec θ = √5, find cos θ 📐", "options": {"A": "1/√5", "B": "2/√5", "C": "√(4/5)", "D": "√(3/5)"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "tan θ = 1/√3, find (1-cos θ)/(1+cos θ) ➗", "options": {"A": "1/3", "B": "1/2", "C": "1/4", "D": "1/5"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "sin θ = 3/5, cos φ = 5/13, both acute. Find sin(θ+φ) ➕", "options": {"A": "56/65", "B": "33/65", "C": "63/65", "D": "16/65"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "cos A = 4/5, cos B = 12/13, find cos(A+B) ➕", "options": {"A": "16/65", "B": "33/65", "C": "56/65", "D": "63/65"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "sin A = 1/√10, sin B = 1/√5, find sin(A+B) 📊", "options": {"A": "1/√2", "B": "√3/2", "C": "1/2", "D": "3/√10"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "tan A = 1/2, tan B = 1/3, find tan(A+B) ➕", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "If 7 sin²θ + 3 cos²θ = 4, find tan θ 🔍", "options": {"A": "1/√3", "B": "√3", "C": "1", "D": "1/2"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "5 tan θ = 4, find (5 sin θ - 3 cos θ)/(5 sin θ + 2 cos θ) ➗", "options": {"A": "1/6", "B": "1/3", "C": "1/2", "D": "2/3"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "sin θ + cos θ = √2 cos θ, find tan θ 🔄", "options": {"A": "√2-1", "B": "√2+1", "C": "1", "D": "√2"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "cot θ + tan θ = 2 cosec θ, find cos θ 📐", "options": {"A": "1/2", "B": "√3/2", "C": "1/√2", "D": "√2/2"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "√3 sin θ = cos θ, find 3 cos²θ + 2 sin²θ 📈", "options": {"A": "2.5", "B": "2.75", "C": "3", "D": "3.25"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "sin θ = m/n, find cos θ + sin θ cot θ ➕", "options": {"A": "n/m", "B": "m/n", "C": "n²/m", "D": "√(n²-m²)/m"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "tan A + sin A = m, tan A - sin A = n, find m² - n² 🔄", "options": {"A": "4√(mn)", "B": "4mn", "C": "2√(mn)", "D": "2mn"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If x = a cos θ, y = b sin θ, find x²/a² + y²/b² ➕", "options": {"A": "0", "B": "1", "C": "cos²θ+sin²θ", "D": "a²+b²"}, "correct_answer": "B"},
    
    {"id": 41, "difficulty": "hard", "question": "sin θ + sin²θ = 1, find cos²θ + cos⁴θ 🔄", "options": {"A": "1", "B": "sin θ", "C": "cos θ", "D": "tan θ"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "tan θ + sec θ = 3, find sin θ 🔍", "options": {"A": "0.6", "B": "0.8", "C": "4/5", "D": "0.75"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "hard", "question": "cot θ + cosec θ = 5, find cos θ 📐", "options": {"A": "12/13", "B": "5/13", "C": "7/25", "D": "24/25"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "If sin θ + cos θ = p, find sin³θ + cos³θ 🔢", "options": {"A": "p(3-p²)/2", "B": "p(p²-1)/2", "C": "(3p-p³)/2", "D": "p³-3p"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "hard", "question": "tan θ + cot θ = 4, find tan²θ + cot²θ 📈", "options": {"A": "14", "B": "16", "C": "18", "D": "20"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "sin θ + cos θ = √2, find tan θ + cot θ ➕", "options": {"A": "1", "B": "2", "C": "√2", "D": "2√2"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "hard", "question": "sec θ - tan θ = 1/3, find sin θ 🔄", "options": {"A": "3/5", "B": "4/5", "C": "5/13", "D": "12/13"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "hard", "question": "cosec θ - cot θ = 1/4, find cos θ 📊", "options": {"A": "15/17", "B": "8/17", "C": "7/25", "D": "24/25"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "If x = sec θ + tan θ, find sec θ - tan θ 🔄", "options": {"A": "1/x", "B": "x", "C": "x²", "D": "√x"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If tan θ + sin θ = a, tan θ - sin θ = b, find a² - b² 📐", "options": {"A": "4ab", "B": "4√(ab)", "C": "2ab", "D": "2√(ab)"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "hard", "question": "sin θ = 2mn/(m²+n²), find cos θ 🔍", "options": {"A": "(m²-n²)/(m²+n²)", "B": "2mn/(m²+n²)", "C": "√(m⁴+n⁴)/(m²+n²)", "D": "|m²-n²|/(m²+n²)"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "hard", "question": "tan A = m/(m-1), tan B = 1/(2m-1), find A+B 📊", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "hard", "question": "If cos θ + sin θ = √2 cos θ, prove tan θ = √2-1, find sec²θ 📈", "options": {"A": "2-√2", "B": "4-2√2", "C": "2√2", "D": "3-2√2"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "sin θ/(1+cos θ) + (1+cos θ)/sin θ = 4, find θ 🔍", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "If tan²θ = 1 - a², find sec θ + tan³θ cosec θ 📐", "options": {"A": "(2-a²)^(3/2)", "B": "(2-a²)", "C": "√(2-a²)", "D": "(2-a²)√(2-a²)"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "sin θ + cos θ = √2 sin(90°-θ), find tan θ 🔄", "options": {"A": "√2+1", "B": "√2-1", "C": "1", "D": "√2"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "hard", "question": "√(1+sin θ)/(1-sin θ) = sec θ + tan θ, find value when θ=60° 📊", "options": {"A": "2+√3", "B": "√3+2", "C": "√3-1", "D": "2√3"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "If (1+sin θ)(1+sin φ)(1+sin ψ) = (1-sin θ)(1-sin φ)(1-sin ψ), find value 🔢", "options": {"A": "0", "B": "1", "C": "cos θ cos φ cos ψ", "D": "±1"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "hard", "question": "tan θ = b/a, find (a cos 2θ + b sin 2θ) 📐", "options": {"A": "a", "B": "b", "C": "a²-b²", "D": "√(a²+b²)"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If tan θ = (1-a²)/(2a), find sin θ + cos θ ➕", "options": {"A": "(1+a²)/(1-a²)", "B": "(1+a)/(1-a)", "C": "√2(1+a²)/|1-a²|", "D": "√2(1+a)/|1-a|"}, "correct_answer": "B"}
  ]
}