ట్రిగోనమెట్రీ – Step-wise Solved Problems (61–80) | Topper Level

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61. sinθ + cosθ = √2 అయితే sinθcosθ = ?
Step 1: Square both sides
( sinθ + cosθ )² = 2
Step 2: 1 + 2sinθcosθ = 2
Step 3: sinθcosθ = 1/2
Final Answer = 1/2

62. sinθ + cosθ = 0 అయితే θ = ?
Step: sinθ = −cosθ ⇒ tanθ = −1
Final Answer = 135°

63. sin²θ + cos²θ + 2sinθcosθ = ?
Step: = (sinθ + cosθ)²
Final Answer = (sinθ + cosθ)²

64. sinθcosθ = 1/4 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2(1/4) = 1 + 1/2
Final Answer = 3/2

65. sin⁴θ + cos⁴θ value?
Step: = 1 − 2sin²θcos²θ
Final Answer = 1 − 2sin²θcos²θ

66. sinθcosθ max అయితే θ = ?
Step: max at 45°
Final Answer = 45°

67. sinθ + cosθ max అయితే θ = ?
Final Answer = 45°

68. sinθcosθ = 1/2 అయితే sinθ + cosθ = ?
Step: occurs at 45°
Final Answer = √2

69. sin²θ − cos²θ = ?
Final Answer = −cos2θ

70. tan²θ + 1 = ?
Final Answer = sec²θ

71. cot²θ + 1 = ?
Final Answer = cosec²θ

72. sec²θ − tan²θ = ?
Final Answer = 1

73. cosec²θ − cot²θ = ?
Final Answer = 1

74. sinθ = 3/5 అయితే cosθ = ?
Step: cosθ = √(1 − 9/25) = 4/5
Final Answer = 4/5

75. cosθ = 5/13 అయితే sinθ = ?
Step: sinθ = √(1 − 25/169) = 12/13
Final Answer = 12/13

76. tanθ = 4/3 అయితే sinθ = ?
Triangle method → sinθ = 4/5
Final Answer = 4/5

77. tanθ = 5 అయితే sinθcosθ = ?
Step: = tanθ / (1 + tan²θ)
= 5 / 26
Final Answer = 5/26

78. sinθcosθ = 1/2 sin2θ → prove
Identity apply
Final Answer = proved

79. sin²θcos²θ max value?
Step: (1/2)² = 1/4
Final Answer = 1/4

80. sinθ + cosθ = 1 అయితే θ = ?
Try θ = 45° → works
Final Answer = 45°

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