Trigonometry
- The value of tan 10° tan 15° tan 75° tan 80° is
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tan10°.tan15°. tan75° . tan80°
= (tan10°. tan80°)(tan15°.tan75°)
= (tan10°. cot10°)(tan15°.cot15°)
= 1 × 1 = 1
[tan(90° – θ) = cotθ; tanθ . cotθ = 1]Correct Option: B
tan10°.tan15°. tan75° . tan80°
= (tan10°. tan80°)(tan15°.tan75°)
= (tan10°. cot10°)(tan15°.cot15°)
= 1 × 1 = 1
[tan(90° – θ) = cotθ; tanθ . cotθ = 1]
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2sin68° - 2cot15° - 3tan45°.tan20°.tan40°.tan50°.tan70° is equal to cos22° 5tan75° 5
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2sin 68° - 2cot 15° - 3tan 45° . tan 20° . tan 40° tan 50° tan 70° cos 22° 5tan 75° 5 = 2 cos22° - 2tan 75° - 3.tan 20° . cot 20°tan 40° . cot 40° cos 22° 5tan 75° 5 = 2 - 2 - 3 5 5 = 10 - 2 - 3 = 1 5 
Correct Option: C
2sin 68° - 2cot 15° - 3tan 45° . tan 20° . tan 40° tan 50° tan 70° cos 22° 5tan 75° 5 = 2 cos22° - 2tan 75° - 3.tan 20° . cot 20°tan 40° . cot 40° cos 22° 5tan 75° 5 = 2 - 2 - 3 5 5 = 10 - 2 - 3 = 1 5
- If α + β = 90°, then the value of (1 – sin²α) (1 – cos²α) × (1 + cot²β) (1 + tan²β) is
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(1 – sin² α) (1 – cos²α) (1 + cot²β) (1 + tan² β)
= cos² α . sin² α . cosec² beta; sec²β
= (cos² α . cosec² β) (sin² α . sec² β)
= (cos² α . sec² α) (sin² &alpha . cosec² α) = 1
[α + β = 90° ⇒ β = 90° – α cosec β = cosec (90° – α)
= sec α ; sec β = sec (90° – α )
= cosec α, sin α . cosec α
= cos α . sec α = 1]Correct Option: A
(1 – sin² α) (1 – cos²α) (1 + cot²β) (1 + tan² β)
= cos² α . sin² α . cosec² beta; sec²β
= (cos² α . cosec² β) (sin² α . sec² β)
= (cos² α . sec² α) (sin² &alpha . cosec² α) = 1
[α + β = 90° ⇒ β = 90° – α cosec β = cosec (90° – α)
= sec α ; sec β = sec (90° – α )
= cosec α, sin α . cosec α
= cos α . sec α = 1]
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The expression tan 57° + cot 37° is equal to tan 33° + cot 53°
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tan 57° + cot 37° tan 33° + cot 53° cot 33° + tan 53° tan 33° + cot 53°
[∵ tan(90° – θ) = cotθ, cot (90° – θ)= tanθ]
= 1 + tan 53° . tan 33° × tan 53° tan 33° . tan 53° + 1 tan 33°
= tan 53° . cot 33°
= cot 37° . tan 57°Correct Option: B
tan 57° + cot 37° tan 33° + cot 53° cot 33° + tan 53° tan 33° + cot 53°
[∵ tan(90° – θ) = cotθ, cot (90° – θ)= tanθ]= 1 + tan 53° . tan 33° × tan 53° tan 33° . tan 53° + 1 tan 33°
= tan 53° . cot 33°
= cot 37° . tan 57°
- If sin 17° = (x / y) , then the value of (sec 17° – sin 73°) is
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sin 17° = x y
cos 17° = √1 - sin² 17°
= √y² - x² y ∴ sec 17° = y √y² - x²
sin 73° = sin (90° – 17°) = cos 17°
∴ sec 17°– sin 73°= y - √y² - x² √y² - x² y = y² - y² + x² = x² y√y² - x² y√y² - x²
Correct Option: B
sin 17° = x y
cos 17° = √1 - sin² 17°= √y² - x² y ∴ sec 17° = y √y² - x²
sin 73° = sin (90° – 17°) = cos 17°
∴ sec 17°– sin 73°= y - √y² - x² √y² - x² y = y² - y² + x² = x² y√y² - x² y√y² - x²
