{
  "title": "Trigonometric Identities - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is the value of sin²θ + cos²θ?", "options": {"A": "0", "B": "1", "C": "tan²θ", "D": "sec²θ"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "What is 1 + tan²θ equal to?", "options": {"A": "sin²θ", "B": "cos²θ", "C": "sec²θ", "D": "cosec²θ"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "What is 1 + cot²θ equal to?", "options": {"A": "sin²θ", "B": "cos²θ", "C": "sec²θ", "D": "cosec²θ"}, "correct_answer": "D"},
    {"id": 4, "difficulty": "easy", "question": "Simplify: sec²θ - tan²θ", "options": {"A": "0", "B": "1", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Simplify: cosec²θ - cot²θ", "options": {"A": "0", "B": "1", "C": "sin²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "What is tan θ in terms of sin θ and cos θ?", "options": {"A": "sin θ/cos θ", "B": "cos θ/sin θ", "C": "1/sin θ", "D": "1/cos θ"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "What is cot θ in terms of sin θ and cos θ?", "options": {"A": "sin θ/cos θ", "B": "cos θ/sin θ", "C": "1/sin θ", "D": "1/cos θ"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "What is sec θ in terms of cos θ?", "options": {"A": "1/sin θ", "B": "1/cos θ", "C": "sin θ/cos θ", "D": "cos θ/sin θ"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "What is cosec θ in terms of sin θ?", "options": {"A": "1/sin θ", "B": "1/cos θ", "C": "sin θ/cos θ", "D": "cos θ/sin θ"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If sin θ = 3/5, what is cos²θ?", "options": {"A": "9/25", "B": "16/25", "C": "25/16", "D": "25/9"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "If tan θ = 4/3, what is sec²θ?", "options": {"A": "16/9", "B": "25/9", "C": "9/16", "D": "9/25"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If cot θ = 5/12, what is cosec²θ?", "options": {"A": "144/169", "B": "169/144", "C": "25/144", "D": "144/25"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Simplify: (1 - sin²θ)sec²θ", "options": {"A": "0", "B": "1", "C": "tan²θ", "D": "cos²θ"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "Simplify: (1 + tan²θ)cos²θ", "options": {"A": "0", "B": "1", "C": "sin²θ", "D": "tan²θ"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Simplify: (cosec²θ - 1)sin²θ", "options": {"A": "0", "B": "1", "C": "cos²θ", "D": "tan²θ"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "Express tan²θ in terms of sin²θ", "options": {"A": "sin²θ/(1-sin²θ)", "B": "sin²θ/(1+sin²θ)", "C": "(1-sin²θ)/sin²θ", "D": "(1+sin²θ)/sin²θ"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Express sin²θ in terms of tan²θ", "options": {"A": "tan²θ/(1+tan²θ)", "B": "1/(1+tan²θ)", "C": "(1+tan²θ)/tan²θ", "D": "1-tan²θ"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If sin θ + cos θ = 1, what is sin θ cos θ?", "options": {"A": "0", "B": "1", "C": "0.5", "D": "2"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "What is sec θ × cos θ?", "options": {"A": "0", "B": "1", "C": "tan θ", "D": "sin θ"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "What is cosec θ × sin θ?", "options": {"A": "0", "B": "1", "C": "cot θ", "D": "cos θ"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Prove that: (1 + cot²θ)(1 - cos²θ) = 1", "options": {"A": "True", "B": "False", "C": "Only for some θ", "D": "Cannot be determined"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "If sin θ + cos θ = √2, find sin θ cos θ", "options": {"A": "0", "B": "0.5", "C": "1", "D": "√2"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "If tan θ + cot θ = 2, find sin θ", "options": {"A": "0", "B": "1/√2", "C": "√3/2", "D": "1"}, "correct_answer": "D"},
    {"id": 24, "difficulty": "medium", "question": "Prove that: √((1+sin θ)/(1-sin θ)) = sec θ + tan θ", "options": {"A": "True for all θ", "B": "False", "C": "True only for acute θ", "D": "True only for obtuse θ"}, "correct_answer": "C"},
    {"id": 25, "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": "a² + b²", "C": "2ab", "D": "0"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1", "options": {"A": "True", "B": "False", "C": "Cannot be determined", "D": "Only for specific θ"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "If 7sin²θ + 3cos²θ = 4, find tan θ", "options": {"A": "1/√3", "B": "√3", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If tan²θ = 1 - e², prove that sec θ + tan³θ cosec θ = (2-e²)^{3/2}", "options": {"A": "True", "B": "False", "C": "Only for e=0", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Prove that: (cosec θ - sin θ)(sec θ - cos θ) = 1/(tan θ + cot θ)", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "If tan θ + sin θ = m and tan θ - sin θ = n, prove that m² - n² = 4√(mn)", "options": {"A": "True", "B": "False", "C": "Only for m=n", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 31, "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": 32, "difficulty": "medium", "question": "Prove that: (1+cot θ-cosec θ)(1+tan θ+sec θ) = 2", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "If a cos θ + b sin θ = m and a sin θ - b cos θ = n, prove that a² + b² = m² + n²", "options": {"A": "True", "B": "False", "C": "Only when a=b", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Prove that: tan A/(1-cot A) + cot A/(1-tan A) = 1 + sec A cosec A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 35, "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": 36, "difficulty": "medium", "question": "Prove that: (sin θ - 2sin³θ)/(2cos³θ - cos θ) = tan θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If tan θ = √(1-a²), find sec θ + tan³θ cosec θ", "options": {"A": "(2-a²)^{3/2}", "B": "(1-a²)^{3/2}", "C": "(2+a²)^{3/2}", "D": "(1+a²)^{3/2}"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Prove that: (1+tan²A)/(1+cot²A) = tan²A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If sin θ + cos θ = p and sec θ + cosec θ = q, eliminate θ", "options": {"A": "p(q-2) = q(p²-1)", "B": "q(p-2) = p(q²-1)", "C": "p+q=2", "D": "pq=2"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Prove that: √((1+cos θ)/(1-cos θ)) = cosec θ + cot θ", "options": {"A": "True for all θ", "B": "False", "C": "True only for acute θ", "D": "True only for obtuse θ"}, "correct_answer": "C"},
    {"id": 41, "difficulty": "medium", "question": "If 3sin θ + 5cos θ = 5, prove that 5sin θ - 3cos θ = ±3", "options": {"A": "True", "B": "False", "C": "Only for specific θ", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "Prove that: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "If cot θ + tan θ = 2 cosec θ, find θ", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Prove that: (sec A + tan A - 1)/(sec A - tan A + 1) = (1 + sin A)/cos A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "If a sin θ + b cos θ = c, prove that a cos θ - b sin θ = ±√(a²+b²-c²)", "options": {"A": "True", "B": "False", "C": "Only when a=b", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Prove that: (1+cot θ+tan θ)(sin θ-cos θ) = sec θ/cosec²θ - cosec θ/sec²θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "If x = a sec θ cos φ, y = b sec θ sin φ, z = c tan θ, find x²/a² + y²/b² - z²/c²", "options": {"A": "0", "B": "1", "C": "sec²θ", "D": "tan²θ"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "medium", "question": "Prove that: sin²A cot²A + cos²A tan²A = 1", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "medium", "question": "If sin θ + sin²θ = 1, prove that cos²θ + cos⁴θ = 1", "options": {"A": "True", "B": "False", "C": "Only for θ=30°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Prove that: tan³θ/(1+tan²θ) + cot³θ/(1+cot²θ) = sec θ cosec θ - 2 sin θ cos θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "If tan A = n tan B and sin A = m sin B, prove that cos²A = (m²-1)/(n²-1)", "options": {"A": "True", "B": "False", "C": "Only when m=n", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Prove that: (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "If sec θ + tan θ = p, prove that sin θ = (p²-1)/(p²+1)", "options": {"A": "True", "B": "False", "C": "Only for p=1", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Prove that: 2(sin⁶θ + cos⁶θ) - 3(sin⁴θ + cos⁴θ) + 1 = 0", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "If x = cos θ + sin θ and y = cos θ - sin θ, prove that x² + y² = 2", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Prove that: (cosec θ - cot θ)² = (1-cos θ)/(1+cos θ)", "options": {"A": "True", "B": "False", "C": "Only for acute θ", "D": "Only for obtuse θ"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "If cos θ + sin θ = √2 cos θ, prove that cos θ - sin θ = √2 sin θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Prove that: (1+sec A)/sec A = sin²A/(1-cos A)", "options": {"A": "True", "B": "False", "C": "Only for A=60°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "If a cos³θ + 3a cos θ sin²θ = m and a sin³θ + 3a cos²θ sin θ = n, find (m+n)^{2/3} + (m-n)^{2/3}", "options": {"A": "2a^{2/3}", "B": "a^{2/3}", "C": "2a", "D": "a"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Prove that: (tan θ + sec θ - 1)/(tan θ - sec θ + 1) = (1+sin θ)/cos θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "If tan²θ = 1 - a², find sec θ + tan³θ cosec θ", "options": {"A": "(2-a²)^{3/2}", "B": "(1-a²)^{3/2}", "C": "(1+a²)^{3/2}", "D": "(2+a²)^{3/2}"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "Prove that: (cos A - sin A + 1)/(cos A + sin A - 1) = cosec A + cot A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "If sin θ + cos θ = m and sec θ + cosec θ = n, prove that n(m²-1) = 2m", "options": {"A": "True", "B": "False", "C": "Only when m=√2", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Prove that: 1/(sec θ - tan θ) = sec θ + tan θ", "options": {"A": "True", "B": "False", "C": "Only for θ=0°", "D": "Only for θ=45°"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "If 5cos θ + 12sin θ = 13, find sin θ", "options": {"A": "5/13", "B": "12/13", "C": "13/5", "D": "13/12"}, "correct_answer": "B"},
    {"id": 66, "difficulty": "medium", "question": "Prove that: (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = 1/(sec θ - tan θ)", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "If cot θ + tan θ = x and sec θ - cos θ = y, prove that (x²y)^{2/3} - (xy²)^{2/3} = 1", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Prove that: sin θ/(1+cos θ) + (1+cos θ)/sin θ = 2cosec θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "If a = sec θ - tan θ and b = cosec θ + cot θ, find a in terms of b", "options": {"A": "(b-1)/(b+1)", "B": "(b+1)/(b-1)", "C": "b/(b²-1)", "D": "(b²-1)/b"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "Prove that: (1+sin θ)/cos θ + cos θ/(1+sin θ) = 2sec θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "If sin θ + cos θ = √2, prove that cot θ = 1", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Prove that: √((1-cos θ)/(1+cos θ)) = cosec θ - cot θ", "options": {"A": "True", "B": "False", "C": "Only for acute θ", "D": "Only for obtuse θ"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "If x = r sin A cos B, y = r sin A sin B, z = r cos A, find x²+y²+z²", "options": {"A": "r", "B": "r²", "C": "r³", "D": "1"}, "correct_answer": "B"},
    {"id": 74, "difficulty": "medium", "question": "Prove that: tan²θ - sin²θ = tan²θ sin²θ", "options": {"A": "True", "B": "False", "C": "Only for θ=45°", "D": "Only for θ=30°"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "If 3sin θ + 4cos θ = 5, find 4sin θ - 3cos θ", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "Prove that: cos⁴A - sin⁴A = cos²A - sin²A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "If sec θ + tan θ = p, find cosec θ + cot θ", "options": {"A": "(p+1)/(p-1)", "B": "(p-1)/(p+1)", "C": "p/(p²-1)", "D": "(p²-1)/p"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "Prove that: (1+cot A - cosec A)(1+tan A + sec A) = 2", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "If sin θ = m/n, find tan θ + sec θ", "options": {"A": "√((n+m)/(n-m))", "B": "√((n-m)/(n+m))", "C": "(n+m)/(n-m)", "D": "(n-m)/(n+m)"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Prove that: sin A/(1+cos A) = (1-cos A)/sin A", "options": {"A": "True", "B": "False", "C": "Only for A=45°", "D": "Only for A=30°"}, "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", "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 sec θ + b tan θ + c = 0 and p sec θ + q tan θ + r = 0, eliminate θ", "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": 85, "difficulty": "hard", "question": "If sin θ = x + 1/x (x is real), which is true?", "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": 86, "difficulty": "hard", "question": "If tan θ = (sin α - cos α)/(sin α + cos α), find sin θ", "options": {"A": "sin α - cos α", "B": "(sin α - cos α)/√2", "C": "sin²α - cos²α", "D": "√(sin²α+cos²α)"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "If √(1+sin θ)/(1-sin θ) = sec θ + tan θ, find the condition", "options": {"A": "θ ∈ [0, π/2)", "B": "θ ∈ (π/2, π]", "C": "θ ∈ [0, π]", "D": "θ ∈ R"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "If (1+sin A)(1+sin B)(1+sin C) = (1-sin A)(1-sin B)(1-sin C), find 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": 89, "difficulty": "hard", "question": "If sin θ + sin²θ + sin³θ = 1, prove 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": 90, "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": 91, "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": 92, "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": 93, "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": 94, "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"},
    {"id": 95, "difficulty": "hard", "question": "If cos⁴α/cos²β + sin⁴α/sin²β = 1, prove cos⁴β/cos²α + sin⁴β/sin²α = 1", "options": {"A": "True", "B": "False", "C": "Only when α=β", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "If tan A = (1-cos B)/sin B, prove tan 2A = tan B", "options": {"A": "True", "B": "False", "C": "Only for B=45°", "D": "Cannot be proven"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "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": 98, "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 value", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "B"},
    {"id": 99, "difficulty": "hard", "question": "If sin θ = (a²-b²)/(a²+b²), find cos θ", "options": {"A": "2ab/(a²+b²)", "B": "(a²-b²)/(2ab)", "C": "(a²+b²)/(2ab)", "D": "2ab/(a²-b²)"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "If cos θ/(1-sin θ) + cos θ/(1+sin θ) = 4, find θ where 0°<θ<90°", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"}
  ]
}