206. If x cosθ – sinθ = 1, then x² + (1 +x² ) sinq equals

1. 1. 2

   
   
   

   2. 1

   
   
   

   3. - 1

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   x cosθ – sinθ = 1
   ⇒ x cosθ = 1 + sinθ
   

   ⇒ x =		1	+	sin θ
   		cos θ		cos θ

   
   ⇒ x = secθ + tanθ --- (i)
   ∵ sec²θ – tan²θ = 1
   ⇒ (secθ + tanθ) (secθ – tanθ) =1
   

   ⇒ secθ – tanθ =	1	(ii)
   	x

   
   From equation (i) + (ii),
   

   2secθ = x +		1	=	x² + 1
   		x		x

   
   

   ⇒ secθ =	x² + 1
   	2x

   
   From equation (i) – (ii),
   

   2tanθ = x –		1	=	x² - 1
   		x		x

   
   

   ∴ tanθ =	x² - 1
   	2x

   
   

   ∴ sinθ =	tanθ
   	secθ

   
   

   =		x² - 1	×	2x	=	x² - 1
   		2x		x² + 1		x² + 1

   
   ∴ Expression = x² – (1 + x² ) sinθ
   

   = x² - (1 + x²) ×	x² - 1	= x² - x² + 1 = 1
   	x² + 1

   
   Note : In the original equation x² + (1 + x² ) sinθ has been given that seems incorrect
   

   Correct Option: B

   x cosθ – sinθ = 1
   ⇒ x cosθ = 1 + sinθ
   

   ⇒ x =		1	+	sin θ
   		cos θ		cos θ

   
   ⇒ x = secθ + tanθ --- (i)
   ∵ sec²θ – tan²θ = 1
   ⇒ (secθ + tanθ) (secθ – tanθ) =1
   

   ⇒ secθ – tanθ =	1	(ii)
   	x

   
   From equation (i) + (ii),
   

   2secθ = x +		1	=	x² + 1
   		x		x

   
   

   ⇒ secθ =	x² + 1
   	2x

   
   From equation (i) – (ii),
   

   2tanθ = x –		1	=	x² - 1
   		x		x

   
   

   ∴ tanθ =	x² - 1
   	2x

   
   

   ∴ sinθ =	tanθ
   	secθ

   
   

   =		x² - 1	×	2x	=	x² - 1
   		2x		x² + 1		x² + 1

   
   ∴ Expression = x² – (1 + x² ) sinθ
   

   = x² - (1 + x²) ×	x² - 1	= x² - x² + 1 = 1
   	x² + 1

   
   Note : In the original equation x² + (1 + x² ) sinθ has been given that seems incorrect
   

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207. If sin θ + cos θ = √2 cos θ, then the value of cot θ is

1. 1. √2 + 1

   
   
   

   2. √2 – 1

   
   
   

   3. √3 – 1

   
   
   

   4. √3 + 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinθ + cosθ =√2 cos θ
   ⇒ √2 cos θ - cos θ= sinθ
   ⇒ cosθ (√2 - 1) = sinθ
   

   ⇒	cosθ	=	1
   	sinθ		√2 - 1

   
   

   =	√2 + 1	= √2 + 1
   	(√2 - 1)(√2 + 1)

   
   cotθ = √2 + 1
   
   

   Correct Option: A

   sinθ + cosθ =√2 cos θ
   ⇒ √2 cos θ - cos θ= sinθ
   ⇒ cosθ (√2 - 1) = sinθ
   

   ⇒	cosθ	=	1
   	sinθ		√2 - 1

   
   

   =	√2 + 1	= √2 + 1
   	(√2 - 1)(√2 + 1)

   
   cotθ = √2 + 1
   
   

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208. If cos⁴θ – sin⁴θ = (2 / 3) , then the value of 1 – 2 sin² θ is

1. 1. 	2
      	3

   
   
   

   2. 	3
      	2

   
   
   

   3. 1

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   cos4θ – sin4θ =	2
   	3

   
   

   ⇒ (cos²θ + sin²θ) (cos²θ– sin²θ) =	2
   	3

   
   [ ∵ cos²θ + sin²θ = 1]
   

   ⇒ cos²θ + sin²θ =	2
   	3

   
   

   ⇒ 1 - sin²θ - sin²θ =	2
   	3

   
   

   ⇒ 1 - 2 sin²θ =	2
   	3

   
   

   Correct Option: A

   cos4θ – sin4θ =	2
   	3

   
   

   ⇒ (cos²θ + sin²θ) (cos²θ– sin²θ) =	2
   	3

   
   [ ∵ cos²θ + sin²θ = 1]
   

   ⇒ cos²θ + sin²θ =	2
   	3

   
   

   ⇒ 1 - sin²θ - sin²θ =	2
   	3

   
   

   ⇒ 1 - 2 sin²θ =	2
   	3

   
   

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209. The value of	cot 30° - cot 75°	is equal to
     	tan 15° - tan 60°

1. 1. - 1

   
   
   

   2. 0

   
   
   

   3. 1

   
   
   

   4. 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   	cot 30° - cot 75°
   	tan 15° - tan 60°

   
   

   =	cot (90° - 60°) - cot (90° - 15°)
   	tan 15° - tan 60°

   
   

   =	cot 60° - tan 15°	= - 1
   	tan 15° - tan 60°

   
   

   Correct Option: A

   	cot 30° - cot 75°
   	tan 15° - tan 60°

   
   

   =	cot (90° - 60°) - cot (90° - 15°)
   	tan 15° - tan 60°

   
   

   =	cot 60° - tan 15°	= - 1
   	tan 15° - tan 60°

   
   

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210. If sin θ + cos θ = p and sec θ + cosec θ = θ, then the value of θ (p² – 1) is

1. 1. 1

   
   
   

   2. p

   
   
   

   3. 2p

   
   
   

   4. 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin θ + cos θ = p
   sec θ + cosec θ = q
   

   ⇒	1	+	1	= q
   	cos θ		sinθ

   
   

   ⇒	sinθ + cosθ	= q
   	sinθ . cosθ

   
   

   ∴ q(p² - 1) =			sin θ + cos θ			((sinθ + cosθ)² - 1)
   			sinθ . cosθ

   
   

   =	sinθ + cosθ	. (sin²θ + cos²θ + 2sinθ.cosθ - 1)
   	sinθ . cosθ

   
   

   =	sinθ + cosθ	. 2sinθ . cosθ
   	sinθ . cosθ

   
   = 2p

   Correct Option: C

   sin θ + cos θ = p
   sec θ + cosec θ = q
   

   ⇒	1	+	1	= q
   	cos θ		sinθ

   
   

   ⇒	sinθ + cosθ	= q
   	sinθ . cosθ

   
   

   ∴ q(p² - 1) =			sin θ + cos θ			((sinθ + cosθ)² - 1)
   			sinθ . cosθ

   
   

   =	sinθ + cosθ	. (sin²θ + cos²θ + 2sinθ.cosθ - 1)
   	sinθ . cosθ

   
   

   =	sinθ + cosθ	. 2sinθ . cosθ
   	sinθ . cosθ

   
   = 2p

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