{
  "title": "Introduction to Trigonometric Ratios - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "In a right triangle, what is the ratio of opposite side to hypotenuse called?", "options": {"A": "cosine", "B": "tangent", "C": "sine", "D": "cotangent"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "In a right triangle ABC, right angled at B, if AB = 3 cm and BC = 4 cm, what is sin C?", "options": {"A": "3/5", "B": "4/5", "C": "3/4", "D": "4/3"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "What is the reciprocal of sine ratio?", "options": {"A": "cosine", "B": "tangent", "C": "cosecant", "D": "secant"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "In a right triangle, tan θ equals:", "options": {"A": "opposite/adjacent", "B": "adjacent/hypotenuse", "C": "opposite/hypotenuse", "D": "adjacent/opposite"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "If sin θ = 0.6, what is cos θ if the triangle is right-angled?", "options": {"A": "0.4", "B": "0.8", "C": "0.6", "D": "0.2"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "What is sec θ in terms of other ratios?", "options": {"A": "1/sin θ", "B": "1/cos θ", "C": "1/tan θ", "D": "sin θ/cos θ"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "In ΔPQR, right angled at Q, if PQ = 5 cm and PR = 13 cm, what is tan R?", "options": {"A": "5/12", "B": "12/5", "C": "5/13", "D": "12/13"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Which trigonometric ratio is equal to cos θ?", "options": {"A": "sin(90°-θ)", "B": "sin(θ)", "C": "tan(90°-θ)", "D": "cos(90°-θ)"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If cot θ = 3/4, what is tan θ?", "options": {"A": "3/4", "B": "4/3", "C": "5/3", "D": "5/4"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "What is the value of sin 0°?", "options": {"A": "0", "B": "1", "C": "√3/2", "D": "1/2"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "In a right triangle, if one acute angle is θ, what is the other acute angle?", "options": {"A": "θ", "B": "90°-θ", "C": "90°+θ", "D": "180°-θ"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "What is cos 90° equal to?", "options": {"A": "0", "B": "1", "C": "√3/2", "D": "1/2"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "If sin A = 4/5 in a right triangle, what is the length of hypotenuse if opposite side is 8 cm?", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "Which ratio represents adjacent/hypotenuse?", "options": {"A": "sine", "B": "cosine", "C": "tangent", "D": "cosecant"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "What is the value of tan 45°?", "options": {"A": "0", "B": "1", "C": "√3", "D": "1/√3"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If cos θ = 0.8, what is sec θ?", "options": {"A": "0.8", "B": "1.25", "C": "0.6", "D": "1.2"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "In ΔXYZ, right angled at Y, if XY = 6 cm and YZ = 8 cm, what is cos Z?", "options": {"A": "6/10", "B": "8/10", "C": "6/8", "D": "8/6"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "What is cosec θ equal to?", "options": {"A": "1/sin θ", "B": "1/cos θ", "C": "sin θ/cos θ", "D": "cos θ/sin θ"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "If tan θ = 1, what is the value of θ in degrees?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "What is sin 30° equal to?", "options": {"A": "1/2", "B": "√3/2", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "In a right triangle ABC, right angled at B, if sin A = 12/13, what is tan C?", "options": {"A": "5/12", "B": "12/5", "C": "5/13", "D": "13/5"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "If 5 tan θ = 4, find the value of (5 sin θ - 3 cos θ)/(5 sin θ + 2 cos θ).", "options": {"A": "1/6", "B": "1/3", "C": "2/5", "D": "3/7"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "In ΔPQR, right angled at Q, if PR + QR = 25 cm and PQ = 5 cm, find sin P.", "options": {"A": "12/13", "B": "5/13", "C": "13/12", "D": "13/5"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If cot θ = 7/8, find the value of (1+sin θ)(1-sin θ)/(1+cos θ)(1-cos θ).", "options": {"A": "49/64", "B": "64/49", "C": "7/8", "D": "8/7"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "If sin θ = 3/5 and θ is acute, find the value of tan θ + sec θ.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "In a right triangle, if tan A = 4/3, find the value of sin A cos A.", "options": {"A": "12/25", "B": "7/25", "C": "24/25", "D": "1/25"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "If 3 cot A = 4, check whether (1-tan²A)/(1+tan²A) = cos²A - sin²A or not.", "options": {"A": "Yes, they are equal", "B": "No, they are not equal", "C": "Cannot be determined", "D": "Only for specific values"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If sin θ + cos θ = √2 cos(90°-θ), find cot θ.", "options": {"A": "√2+1", "B": "√2-1", "C": "0", "D": "1"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "If sec θ + tan θ = p, find cosec θ.", "options": {"A": "(p²+1)/(p²-1)", "B": "(p²+1)/(2p)", "C": "(p²-1)/(p²+1)", "D": "(2p)/(p²+1)"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "In ΔABC, right angled at B, if tan A = 1/√3, find sin A cos C + cos A sin C.", "options": {"A": "0", "B": "1", "C": "√3/2", "D": "1/2"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "If sin(A+B) = 1 and cos(A-B) = √3/2, find A and B.", "options": {"A": "A=45°, B=15°", "B": "A=60°, B=30°", "C": "A=30°, B=60°", "D": "A=15°, B=45°"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "If 7 sin²θ + 3 cos²θ = 4, find tan θ.", "options": {"A": "1/√3", "B": "√3", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "If x = a sec θ + b tan θ and y = a tan θ + b sec θ, find x² - y².", "options": {"A": "a²-b²", "B": "b²-a²", "C": "a²+b²", "D": "2ab"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If tan θ + sin θ = m and tan θ - sin θ = n, find m² - n².", "options": {"A": "4√(mn)", "B": "2√(mn)", "C": "√(mn)", "D": "4mn"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "If cos θ + sin θ = √2 cos θ, show that cos θ - sin θ = √2 sin θ. What is tan θ?", "options": {"A": "√2-1", "B": "√2+1", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If 2 sin θ = √3, find the value of θ where 0° ≤ θ ≤ 90°.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "If √3 tan θ = 1, and θ is acute, find sin²θ - cos²θ.", "options": {"A": "-1/2", "B": "1/2", "C": "√3/2", "D": "-√3/2"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If tan θ = a/b, find (a sin θ - b cos θ)/(a sin θ + b cos θ).", "options": {"A": "(a²-b²)/(a²+b²)", "B": "0", "C": "1", "D": "(b²-a²)/(a²+b²)"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If sec θ + tan θ = p, find the value of sin θ in terms of p.", "options": {"A": "(p²-1)/(p²+1)", "B": "(2p)/(p²+1)", "C": "(p²+1)/(2p)", "D": "(p²-1)/(2p)"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If sin θ = 3/5, find the value of (tan θ + sec θ)².", "options": {"A": "4", "B": "9", "C": "16", "D": "25"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "If 5 tan θ = 4, find (5 sin θ - 3 cos θ)/(4 cos θ + sin θ).", "options": {"A": "1/4", "B": "1/2", "C": "3/4", "D": "1"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "If sin θ + cos θ = √3, find tan θ + cot θ.", "options": {"A": "1", "B": "2", "C": "√3", "D": "1/√3"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "If sec θ = 13/12, find (sin θ - cos θ)/(sin θ + cos θ).", "options": {"A": "1/5", "B": "1/7", "C": "1/13", "D": "1/25"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ - 3 cos θ = ±3. What is sin θ?", "options": {"A": "0 or 4/5", "B": "1/2 or 3/5", "C": "3/5 or 4/5", "D": "0 or 3/5"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "If tan²θ = 1 - a², find sec θ + tan³θ cosec θ.", "options": {"A": "(2-a²)^(3/2)", "B": "(2+a²)^(3/2)", "C": "(1-a²)^(3/2)", "D": "(1+a²)^(3/2)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "If x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, find x² + y² + z².", "options": {"A": "r", "B": "r²", "C": "r³", "D": "1"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "If cos θ/(1+sin θ) + (1+sin θ)/cos θ = 4, find θ where 0° < θ < 90°.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "If tan θ + cot θ = 2, find sin θ.", "options": {"A": "1/√2", "B": "√3/2", "C": "1/2", "D": "1"}, "correct_answer": "D"},
    {"id": 49, "difficulty": "medium", "question": "If sin θ + cos θ = √2, find tan θ + cot θ.", "options": {"A": "1", "B": "2", "C": "√2", "D": "1/√2"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "If 4 sin²θ - 1 = 0, find θ if 0° ≤ θ ≤ 90°.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "If √3 sin θ = cos θ, find θ if 0° ≤ θ ≤ 90°.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "If sin θ = 1/2 and cos φ = 1/2, find θ + φ if both are acute angles.", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "180°"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "If tan θ = √3, find sin θ cos θ.", "options": {"A": "√3/4", "B": "3/4", "C": "1/4", "D": "√3/2"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "If 3 tan θ = √3, find sin²θ - cos²θ.", "options": {"A": "-1/2", "B": "1/2", "C": "√3/2", "D": "-√3/2"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "If sin θ = 0.6, find cos θ + tan θ.", "options": {"A": "1.3", "B": "1.4", "C": "1.5", "D": "1.6"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "If 4 tan θ = 3, find (4 sin θ - cos θ)/(4 sin θ + cos θ).", "options": {"A": "1/2", "B": "1/3", "C": "2/3", "D": "3/4"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "If sin θ = 12/13, find (tan θ + sec θ)².", "options": {"A": "9", "B": "16", "C": "25", "D": "36"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "medium", "question": "If cosec θ = 5/4, find tan²θ + sin²θ.", "options": {"A": "369/400", "B": "481/400", "C": "1", "D": "169/400"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "medium", "question": "If sec θ + tan θ = 3, find sin θ.", "options": {"A": "4/5", "B": "3/5", "C": "2/5", "D": "1/5"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "If 7 cosec θ - 3 cot θ = 7, prove that 7 cot θ - 3 cosec θ = 3. What is sin θ?", "options": {"A": "7/25", "B": "24/25", "C": "7/24", "D": "24/7"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "If tan θ = 1/√7, find (cosec²θ - sec²θ)/(cosec²θ + sec²θ).", "options": {"A": "3/4", "B": "4/3", "C": "2/3", "D": "3/2"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "If sin θ + sin²θ = 1, find cos²θ + cos⁴θ.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "If sin θ = (a² - b²)/(a² + b²), find tan θ.", "options": {"A": "(a²-b²)/(2ab)", "B": "2ab/(a²+b²)", "C": "(a²-b²)/(a²+b²)", "D": "2ab/(a²-b²)"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "If tan A = 1/2 and tan B = 1/3, find tan(A+B).", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "If x sin³θ + y cos³θ = sin θ cos θ and x sin θ = y cos θ, find x² + y².", "options": {"A": "0", "B": "1", "C": "2", "D": "sin²θ+cos²θ"}, "correct_answer": "B"},
    {"id": 66, "difficulty": "medium", "question": "If sec θ + tan θ = p, find the value of sec θ - tan θ.", "options": {"A": "1/p", "B": "p", "C": "p²", "D": "√p"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "If 2 sin θ - cos θ = 0, find tan θ.", "options": {"A": "1/2", "B": "2", "C": "√2", "D": "1/√2"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "If tan θ = 3/4, find (3 sin θ + 2 cos θ)/(3 sin θ - 2 cos θ).", "options": {"A": "3", "B": "9", "C": "6", "D": "12"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "If sin θ + cos θ = √2 sin(90°-θ), find cot θ.", "options": {"A": "√2+1", "B": "√2-1", "C": "1", "D": "√2"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "If a cos θ + b sin θ = m and a sin θ - b cos θ = n, find a² + b².", "options": {"A": "m² + n²", "B": "m² - n²", "C": "n² - m²", "D": "√(m²+n²)"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "If sin θ + cosec θ = 2, find sin¹⁰θ + cosec¹⁰θ.", "options": {"A": "2", "B": "10", "C": "1024", "D": "100"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "If tan A = 1/2, tan B = 1/3, and tan C = 1/4, find tan(A+B+C).", "options": {"A": "3/2", "B": "2/3", "C": "1", "D": "3/4"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "If sin θ = 3/5 and cos φ = 12/13, where θ and φ are acute, find sin(θ+φ).", "options": {"A": "56/65", "B": "33/65", "C": "63/65", "D": "16/65"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "If sec θ - tan θ = 1/3, find sec θ + tan θ.", "options": {"A": "3", "B": "1/3", "C": "9", "D": "1/9"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "If 5 cos θ + 12 sin θ = 13, find tan θ.", "options": {"A": "12/5", "B": "5/12", "C": "13/5", "D": "13/12"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "If tan θ = -1/√3 and θ is in quadrant II, find sin θ.", "options": {"A": "√3/2", "B": "1/2", "C": "-1/2", "D": "-√3/2"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "If sin θ = 5/13 and θ is in quadrant II, find tan θ.", "options": {"A": "-5/12", "B": "5/12", "C": "-12/5", "D": "12/5"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "If 3 sin θ + 4 cos θ = 5, find 4 sin θ - 3 cos θ.", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "If tan θ = p/q, find (p sin θ - q cos θ)/(p sin θ + q cos θ).", "options": {"A": "(p²-q²)/(p²+q²)", "B": "0", "C": "1", "D": "(q²-p²)/(p²+q²)"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "If x = tan A + sin A and y = tan A - sin A, find x² - y².", "options": {"A": "4√(xy)", "B": "2√(xy)", "C": "4xy", "D": "√(xy)"}, "correct_answer": "A"},
    {"id": 81, "difficulty": "hard", "question": "If (1+sin θ)(1+cos θ) = 5/4 and (1-sin θ)(1-cos θ) = m - √n, find m+n where m,n are integers.", "options": {"A": "3", "B": "5", "C": "7", "D": "9"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "If sin θ + sin²θ = 1, find cos¹²θ + 3cos¹⁰θ + 3cos⁸θ + cos⁶θ.", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "If tan θ + sec θ = (√3+1)/2, find sin θ.", "options": {"A": "√3/2", "B": "1/2", "C": "(√3-1)/2√2", "D": "(√3+1)/2√2"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "If a sin θ + b cos θ = c, prove that a cos θ - b sin θ = ±√(a²+b²-c²). Find the maximum value of c.", "options": {"A": "√(a²+b²)", "B": "a²+b²", "C": "a+b", "D": "√(a²+b²)/2"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "If sec θ + tan θ = p, find the value of cosec θ + cot θ in terms of p.", "options": {"A": "(p+1)/(p-1)", "B": "(p²+1)/(p²-1)", "C": "(p-1)/(p+1)", "D": "p/(p²-1)"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If x = sec θ + tan θ and y = cosec θ + cot θ, express y in terms of x.", "options": {"A": "(x+1)/(x-1)", "B": "(x-1)/(x+1)", "C": "x/(x²-1)", "D": "(x²+1)/(x²-1)"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "If tan²α = 1 + 2tan²β, prove that cos²β = 2cos²α. Find sin²α in terms of sin²β.", "options": {"A": "(1+sin²β)/2", "B": "(1-sin²β)/2", "C": "2sin²β/(1+sin²β)", "D": "sin²β/(2-sin²β)"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "If sin θ + cos θ = m and sec θ + cosec θ = n, eliminate θ to find relation between m and n.", "options": {"A": "n(m²-1)=2m", "B": "m(n²-1)=2n", "C": "mn=2", "D": "m+n=2"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "If (sec A + tan A)(sec B + tan B)(sec C + tan C) = (sec A - tan A)(sec B - tan B)(sec C - tan C), find the value.", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "If sin θ = x + 1/x, which of the following is true? (Assume x is real)", "options": {"A": "|x+1/x| ≤ 2", "B": "|x+1/x| ≥ 2", "C": "x+1/x is always positive", "D": "x must be 1"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "If tan θ = (sin α - cos α)/(sin α + cos α), find sin θ in terms of α.", "options": {"A": "sin α - cos α", "B": "(sin α - cos α)/√2", "C": "sin²α - cos²α", "D": "√(sin²α+cos²α)"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "If √(1+sin θ)/(1-sin θ) = sec θ + tan θ, find the condition for equality.", "options": {"A": "θ ∈ [0, π/2)", "B": "θ ∈ (π/2, π]", "C": "θ ∈ [0, π]", "D": "θ ∈ R"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "If a sec θ + b tan θ + c = 0 and p sec θ + q tan θ + r = 0, eliminate θ to find relation between constants.", "options": {"A": "(aq-bp)² = (ar-cp)² - (br-cq)²", "B": "(aq-bp)² = (ar-cp)² + (br-cq)²", "C": "(aq+bp)² = (ar+cp)² - (br+cq)²", "D": "(aq-bp) = (ar-cp)(br-cq)"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "If (1+sin A)(1+sin B)(1+sin C) = (1-sin A)(1-sin B)(1-sin C), find the value of the product.", "options": {"A": "0", "B": "1", "C": "cos A cos B cos C", "D": "sin A sin B sin C"}, "correct_answer": "C"},
    {"id": 95, "difficulty": "hard", "question": "If sin θ + sin²θ + sin³θ = 1, prove that cos⁶θ - 4cos⁴θ + 8cos²θ = 4. Find cos²θ.", "options": {"A": "√5-1/2", "B": "√5+1/2", "C": "3-√5/2", "D": "3+√5/2"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "If tan θ = √(1-e²) sin φ/(1+e cos φ) and tan θ' = √(1-e²) sin φ/(1-e cos φ), find tan θ tan θ'.", "options": {"A": "(1-e²)sin²φ/(1-e²cos²φ)", "B": "(1-e²)sin²φ/(1+e²cos²φ)", "C": "(1-e)sin²φ/(1+ecos²φ)", "D": "(1+e)sin²φ/(1-ecos²φ)"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "If x = r cos θ cos φ, y = r cos θ sin φ, z = r sin θ, find (x²+y²)/z².", "options": {"A": "cot²θ", "B": "tan²θ", "C": "cot²φ", "D": "tan²φ"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "If sin θ = 2mn/(m²+n²), find tan θ.", "options": {"A": "2mn/(m²-n²)", "B": "(m²-n²)/(2mn)", "C": "2mn/(m²+n²)", "D": "(m²+n²)/(2mn)"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "If sec θ + tan θ = p/q, find sin θ.", "options": {"A": "(p²-q²)/(p²+q²)", "B": "2pq/(p²+q²)", "C": "(p²+q²)/(2pq)", "D": "(p²-q²)/(2pq)"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "If a cos θ + b sin θ = m and a sin θ - b cos θ = n, find θ.", "options": {"A": "tan⁻¹((am-bn)/(an+bm))", "B": "tan⁻¹((am+bn)/(an-bm))", "C": "tan⁻¹((an+bm)/(am-bn))", "D": "tan⁻¹((an-bm)/(am+bn))"}, "correct_answer": "A"}
  ]
}