Trigonometry
- If (sinα + cosecα)² + (cosα + secα)² = k + tan²α + cot²α, then the value of k is
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(sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α
⇒ sin²α + cosec²α + 2 sin α . cosec α + cos²α + sec²α + 2 cos α.sec α = k + tan²α + cot²α
⇒ sin²α + cos²α + 2 + cosec²α + sec²α + 2 = k + tan²α + cot²α
⇒ 5 + cosec²α + sec²α = k + tan²α + cot²α
⇒ 5 + 1 + cot²α + 1 + tan²α = k + tan²α + cot²α
⇒ 7 + cot²α + tan²α = k + tan²α + cot²α
⇒ k = 7Correct Option: B
(sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α
⇒ sin²α + cosec²α + 2 sin α . cosec α + cos²α + sec²α + 2 cos α.sec α = k + tan²α + cot²α
⇒ sin²α + cos²α + 2 + cosec²α + sec²α + 2 = k + tan²α + cot²α
⇒ 5 + cosec²α + sec²α = k + tan²α + cot²α
⇒ 5 + 1 + cot²α + 1 + tan²α = k + tan²α + cot²α
⇒ 7 + cot²α + tan²α = k + tan²α + cot²α
⇒ k = 7
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If sin 21° = x , then sec 21° – sin 69° is equal to y
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sin 21° = x y
cos 21° = √1 - sin² 21°
∴ sec 21° = y √y² - x²
∴ sec 21° – sin 69°
= sec 21° – sin (90° – 21°)
= sec 21° – cos 21°= y - √y² - x² √y² - x² y = y² - (y² - x²) - x² y√y² - x² y√y² - x²
Correct Option: A
sin 21° = x y
cos 21° = √1 - sin² 21°∴ sec 21° = y √y² - x²
∴ sec 21° – sin 69°
= sec 21° – sin (90° – 21°)
= sec 21° – cos 21°= y - √y² - x² √y² - x² y = y² - (y² - x²) - x² y√y² - x² y√y² - x²
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If sin πx = x² – 2x + 2, then the value of x is 2
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sin πx = x² - 2x + 2 2
Putting x = 1sin π = 1 - 2 + 2 = 1 2 Correct Option: B
sin πx = x² - 2x + 2 2
Putting x = 1sin π = 1 - 2 + 2 = 1 2
- If tanθ – cotθ = 0, find the value of sinθ + cos θ.
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tanθ – cotθ = 0
⇒ tanθ = cotθ = tan (90° – q)
⇒ θ = 90° – θ ⇒ 2θ = 90°
⇒ θ = 45°
∴ sinθ + cosθ = sin 45° + cos 45°= 1 + 1 = 2 = √2 √2 √2 √2
Correct Option: C
tanθ – cotθ = 0
⇒ tanθ = cotθ = tan (90° – q)
⇒ θ = 90° – θ ⇒ 2θ = 90°
⇒ θ = 45°
∴ sinθ + cosθ = sin 45° + cos 45°= 1 + 1 = 2 = √2 √2 √2 √2
- The greatest value of sin4θ + cos4θ is
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cos4θ + sin4θ = (cos²θ + sin²θ)² – 2cos²θ sin²θ
From maximum value, 2sin²θ.cos²θ = 0
Hence, sin4θ + cos4θ = (1)² – 0 = 1Correct Option: D
cos4θ + sin4θ = (cos²θ + sin²θ)² – 2cos²θ sin²θ
From maximum value, 2sin²θ.cos²θ = 0
Hence, sin4θ + cos4θ = (1)² – 0 = 1
