Trigonometry
- If tan (a – b) = 1, sec (a + b) = 2/√3 and α, β are positive, then the smallest value of a is :
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tan (α – β) = 1 = tan 45°
⇒ α – β = 45° ..... (i)sec (α + β) = 2 = sec 30° √3
⇒ α + β = 30° ..... (ii)
On adding (i) and (ii),
2α = 45° + 30° = 75°⇒ a = 75 = 37 1° 2 2 Correct Option: D
tan (α – β) = 1 = tan 45°
⇒ α – β = 45° ..... (i)sec (α + β) = 2 = sec 30° √3
⇒ α + β = 30° ..... (ii)
On adding (i) and (ii),
2α = 45° + 30° = 75°⇒ a = 75 = 37 1° 2 2
- If tanθ + cotθ = 2, then the value of (tannθ + cotnθ) is :
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tanθ + cotθ = 2
⇒ tanθ + 1 = 2 tanθ ⇒ tan²θ + 1 = 2 tanθ
→ tan²θ + 1 = 2 tanθ
⇒ tan²θ – 2 tanθ + 1 = 0
⇒ (tanθ – 1)2 = 0
⇒ tanθ – 1 = 0
⇒ tanq = 1
∴ cotθ = 1
∴ tannθ + cotnθ = 1 + 1 = 2Correct Option: D
tanθ + cotθ = 2
⇒ tanθ + 1 = 2 tanθ ⇒ tan²θ + 1 = 2 tanθ
→ tan²θ + 1 = 2 tanθ
⇒ tan²θ – 2 tanθ + 1 = 0
⇒ (tanθ – 1)2 = 0
⇒ tanθ – 1 = 0
⇒ tanq = 1
∴ cotθ = 1
∴ tannθ + cotnθ = 1 + 1 = 2
- If cosx = siny and cot ( x – 40°) = tan (50° – y), then the values of x and y are :
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cosx = siny
⇒ cosx = cos (90° – y)
⇒ x = 90° – y
[∵ cos (90° – θ = sinθ)]
⇒ x + y = 90° .... (i)
Again,
cot (x – 40°) = tan (50° – y)
⇒ (cot (x – 40°) = cot {90° – (50° – y)}
[∵ cot (90° – θ) = tanθ]
⇒ x – 40° = 90° – 50° + y
⇒ x – 40 = 40 + y (i)
⇒ x – y = 40 + 40 = 80°
(ii) On adding equations (i) and (ii),
x + y + x – y = 90° + 80°
⇒ 2x = 170°⇒ x = 170° = 85° 2
From equation (i),
⇒ 85° + y = 90°
⇒ y = 90° – 85° = 5°Correct Option: C
cosx = siny
⇒ cosx = cos (90° – y)
⇒ x = 90° – y
[∵ cos (90° – θ = sinθ)]
⇒ x + y = 90° .... (i)
Again,
cot (x – 40°) = tan (50° – y)
⇒ (cot (x – 40°) = cot {90° – (50° – y)}
[∵ cot (90° – θ) = tanθ]
⇒ x – 40° = 90° – 50° + y
⇒ x – 40 = 40 + y (i)
⇒ x – y = 40 + 40 = 80°
(ii) On adding equations (i) and (ii),
x + y + x – y = 90° + 80°
⇒ 2x = 170°⇒ x = 170° = 85° 2
From equation (i),
⇒ 85° + y = 90°
⇒ y = 90° – 85° = 5°
- The value of cosec²60°+ sec²60° – cot²60° + tan²30° will be
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cosec²60° + sec²60° – cot²60° + tan²30°
= 
2 
² + (2)² - 1 ² + 1 ² √3 √3 √3 = 4 + 4 = 4 + 12 = 16 = 5 1 3 3 3 3 Correct Option: C
cosec²60° + sec²60° – cot²60° + tan²30°
= 2 ² + (2)² - 1 ² + 1 ² √3 √3 √3 = 4 + 4 = 4 + 12 = 16 = 5 1 3 3 3 3
- If sinθ + cosecθ = 2, the value of sinnq + cosecnθ is :
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sinθ + cosecθ = 2
⇒ sinθ + 1 = 2 sinθ ⇒ sinθ + 1 = 2 sinθ
⇒ sin²θ + 1 = 2 sinθ
⇒ sin²θ – 2sinθ + 1 = 0
⇒ (sinθ – 1)2 = 0
⇒ sinθ – 1 = 0
⇒ sinθ – 1
∴ cosecθ = 1
∴ sinnθ + cosecnθ = 1 + 1 = 2Correct Option: C
sinθ + cosecθ = 2
⇒ sinθ + 1 = 2 sinθ ⇒ sinθ + 1 = 2 sinθ
⇒ sin²θ + 1 = 2 sinθ
⇒ sin²θ – 2sinθ + 1 = 0
⇒ (sinθ – 1)2 = 0
⇒ sinθ – 1 = 0
⇒ sinθ – 1
∴ cosecθ = 1
∴ sinnθ + cosecnθ = 1 + 1 = 2
