Trigonometry
- If 2 (cos²θ– sin²θ) = 1 (θ is a positive acute angle), then cot θ is equal to
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2(cos² θ - sin² θ) = 1
⇒ cos² θ - (1 - cos² θ) = 1 2 ⇒ 2 cos² θ = 1 + 1 = 3 2 2 ⇒ cos² θ = 3 4 ⇒ sec² θ = 4 3 ⇒ 1 + tan² θ = 4 3 ⇒ tan² θ = 4 - 1 = 1 3 3 ⇒ tan² θ = 1 ⇒ cot θ = √3 √3
Correct Option: D
2(cos² θ - sin² θ) = 1
⇒ cos² θ - (1 - cos² θ) = 1 2 ⇒ 2 cos² θ = 1 + 1 = 3 2 2 ⇒ cos² θ = 3 4 ⇒ sec² θ = 4 3 ⇒ 1 + tan² θ = 4 3 ⇒ tan² θ = 4 - 1 = 1 3 3 ⇒ tan² θ = 1 ⇒ cot θ = √3 √3
- The product cos1° cos2° cos3° cos4° .... cos100° is equal to
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cos1°.cos2°.cos3° ... cos90°....cos100° = 0 [cos90° = 0]
Correct Option: D
cos1°.cos2°.cos3° ... cos90°....cos100° = 0 [cos90° = 0]
- If tan²α = 1 + 2 tan²β (α, β are positive acute angles), then √2 cosα – cosβ is equal to
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tan⊃ α = 1 + 2 tan⊃β
⇒ sec⊃ α – 1 = 1 + 2(sec⊃β – 1)
⇒ sec⊃ α –1 = 2 sec⊃β – 1⇒ 1 = 2 cos² α cos² β
⇒ √2 cosα = cosβ
∴ √2 cosα - cosβ = 0Correct Option: A
tan⊃ α = 1 + 2 tan⊃β
⇒ sec⊃ α – 1 = 1 + 2(sec⊃β – 1)
⇒ sec⊃ α –1 = 2 sec⊃β – 1⇒ 1 = 2 cos² α cos² β
⇒ √2 cosα = cosβ
∴ √2 cosα - cosβ = 0
- If sin 7x = cos 11x , then the value of tan 9x + cot 9x is
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sin 7x = cos 11x
= sin (90° – 11x)
⇒ 7x = 90° – 11x
⇒ 18x = 90°
⇒ x = 5°
∴ tan 9x + cot 9x
= tan45° + cot45°
= 1 + 1 = 2Correct Option: B
sin 7x = cos 11x
= sin (90° – 11x)
⇒ 7x = 90° – 11x
⇒ 18x = 90°
⇒ x = 5°
∴ tan 9x + cot 9x
= tan45° + cot45°
= 1 + 1 = 2
- The minimum value of 4 tan²θ + 9 cot²θ is equal to
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4 tan⊃θ + 9 cot⊃θ
= (2tanθ – 3cotθ)⊃ + 2 × 3 × 2
∴ Minimum value = 12
[∵ (2tanθ – 3cotθ)⊃ ≥ 0]Correct Option: C
4 tan⊃θ + 9 cot⊃θ
= (2tanθ – 3cotθ)⊃ + 2 × 3 × 2
∴ Minimum value = 12
[∵ (2tanθ – 3cotθ)⊃ ≥ 0]
