Trigonometry
- If sin θ + cosec θ = 2, then the value of sin9θ + cosec9θ is :
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sin θ + cosec θ = 2
⇒ sin θ = 1 = 2 sin θ
⇒ sin² θ - 2 sinθ + 1 = 0
⇒ (sin θ - 1)² = 0
⇒ sin θ = 1
∴ cosec θ = 1
∴ sin9 θ + cosec9 θ = 1 + 1 = 2Correct Option: B
sin θ + cosec θ = 2
⇒ sin θ = 1 = 2 sin θ
⇒ sin² θ - 2 sinθ + 1 = 0
⇒ (sin θ - 1)² = 0
⇒ sin θ = 1
∴ cosec θ = 1
∴ sin9 θ + cosec9 θ = 1 + 1 = 2
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tan θ + cot θ is equal to 1 - cot θ 1 - tan θ
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Expression
= tan θ + cot θ 1 - cot θ 1 - tan θ = tan θ + (1 / tan θ) {1 - (1 / tan θ) } 1 - tan θ = tan² θ + 1 tan θ - 1 tan θ(1 - tan θ) = tan² θ + 1 tan θ - 1 tan θ(tan θ - 1) = tan3 θ - 1 tan θ (tan θ - 1) = (tan θ - 1)(tan ²θ + tan θ + 1) tan θ (tan θ - 1) = tan ²θ + tan θ + 1 tan θ
= tan θ + cot θ + 1Correct Option: D
Expression
= tan θ + cot θ 1 - cot θ 1 - tan θ = tan θ + (1 / tan θ) {1 - (1 / tan θ) } 1 - tan θ = tan² θ + 1 tan θ - 1 tan θ(1 - tan θ) = tan² θ + 1 tan θ - 1 tan θ(tan θ - 1) = tan3 θ - 1 tan θ (tan θ - 1) = (tan θ - 1)(tan ²θ + tan θ + 1) tan θ (tan θ - 1) = tan ²θ + tan θ + 1 tan θ
= tan θ + cot θ + 1
- If tan θ + cot θ = 2, then the value of tan100θ + cot100θ is
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tan θ + cot θ = 2
⇒ tan θ + 1 = 2 tan θ
⇒ tan²θ + 1 = 2 tanθ
⇒ tan²θ – 2tan θ + 1 = 0
⇒ (tanθ – 1)² = 0
⇒ tan θ = 1∴ cot θ = 1 = 1 tan θ
∴ tan100 θ + cot100 θ = 1 + 1 = 2
Correct Option: A
tan θ + cot θ = 2
⇒ tan θ + 1 = 2 tan θ
⇒ tan²θ + 1 = 2 tanθ
⇒ tan²θ – 2tan θ + 1 = 0
⇒ (tanθ – 1)² = 0
⇒ tan θ = 1∴ cot θ = 1 = 1 tan θ
∴ tan100 θ + cot100 θ = 1 + 1 = 2
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If tan θ = 2, then the value of 8sinθ + 5cosθ is : sin3θ + 2cos3θ + 3 cosθ
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Expression
= 8 sin θ + 5 cos θ sin3θ + 2cos3θ + 3cos θ
Dividing numerator and denominator by cos θ,= 8 tan θ + 5 tan θ . sin²θ + 2cos²θ + 3 = 8 tan θ + 5 2sin² θ + 2cos²θ + 3 = 8 tan θ + 5 2(sin² θ + 2cos²θ) + 3 = 8 × 2 + 5 = 21 5 5
Correct Option: A
Expression
= 8 sin θ + 5 cos θ sin3θ + 2cos3θ + 3cos θ
Dividing numerator and denominator by cos θ,= 8 tan θ + 5 tan θ . sin²θ + 2cos²θ + 3 = 8 tan θ + 5 2sin² θ + 2cos²θ + 3 = 8 tan θ + 5 2(sin² θ + 2cos²θ) + 3 = 8 × 2 + 5 = 21 5 5
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If 2sinθ - cosθ = 1, then value of cot θ is : cosecθ + sinθ
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2sin θ - cos θ = 1 cos θ - sin θ
Dividing numerator and denominator by sin θ,2 - cot θ = 1 cot θ + 1
⇒ 2 – cot θ = cot θ + 1
⇒ 2 cot θ = 1⇒ cot θ = 1 2
Correct Option: A
2sin θ - cos θ = 1 cos θ - sin θ
Dividing numerator and denominator by sin θ,2 - cot θ = 1 cot θ + 1
⇒ 2 – cot θ = cot θ + 1
⇒ 2 cot θ = 1⇒ cot θ = 1 2
