456. If π sinθ = 1, π cosθ = 1, then the value of		√3tan		2 θ		+ 1		is
     				3

     
     

1. 1. 1
      

   
   
   

   2. √3
      

   
   
   

   3. 2
      

   
   
   

   4. 1

      √3

   
   
   

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   πsin θ = 1 and πcos θ = 1
   

   ∴	πsin θ	= 1
   	πcos θ

   
   ∴ tan θ = 1 = tan 45°
   ⇒ θ = 45°
   

   ∴ √3tan		2 θ		+ 1
   		3

   
   

   = √3tan		2 × 45°		+ 1 = √3 tan30° + 1
   		3

   
   

   Required answer = √3 ×	1	+ 1 = 1 + 1 = 2
   	√3

   
   

   Correct Option: C

   πsin θ = 1 and πcos θ = 1
   

   ∴	πsin θ	= 1
   	πcos θ

   
   ∴ tan θ = 1 = tan 45°
   ⇒ θ = 45°
   

   ∴ √3tan		2 θ		+ 1
   		3

   
   

   = √3tan		2 × 45°		+ 1 = √3 tan30° + 1
   		3

   
   

   Required answer = √3 ×	1	+ 1 = 1 + 1 = 2
   	√3

   
   

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457. Find the value of	1	+	1	.
     	1 + tan2θ		1 + cot2θ

     
     

1. 1. 1

      4

      
      

   
   
   

   2. 1
      

   
   
   

   3. 1

      2

      
      

   
   
   

   4. 2

   
   
   

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   Expression =	1	+	1
   	1 + tan2θ		1 + cot2θ

   
   

   Expression =	1	+	1
   	sec2θ		cosec2θ

   
   [ ∵ sec²θ - tan²θ = 1 , cosec²θ - cot²θ = 1 ]
   Expression = cos²θ + sin²θ = 1
   [ ∵ cosθ . secθ = 1 , sinθ . cosecθ = 1 ]
   

   Correct Option: B

   Expression =	1	+	1
   	1 + tan2θ		1 + cot2θ

   
   

   Expression =	1	+	1
   	sec2θ		cosec2θ

   
   [ ∵ sec²θ - tan²θ = 1 , cosec²θ - cot²θ = 1 ]
   Expression = cos²θ + sin²θ = 1
   [ ∵ cosθ . secθ = 1 , sinθ . cosecθ = 1 ]
   

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458. If tanθ +	1	= 2 , then the value of tan2θ +	1	is equal to :
     	tanθ		tan2θ

     
     

1. 1. 6
      

   
   
   

   2. 4
      

   
   
   

   3. 2
      

   
   
   

   4. 3

   
   
   

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   tanθ +	1	= 2
   	tanθ

   
   ⇒ tan²θ + 1 = 2tanθ
   ⇒ tan²θ - 2tanθ + 1 = 0
   ⇒ ( tanθ - 1 )² = 0 ⇒ tanθ - 1 = 0
   ⇒ tanθ = 1
   

   ∴ tan2θ +	1	= 1 + 1 = 2
   	tan2θ

   Correct Option: C

   tanθ +	1	= 2
   	tanθ

   
   ⇒ tan²θ + 1 = 2tanθ
   ⇒ tan²θ - 2tanθ + 1 = 0
   ⇒ ( tanθ - 1 )² = 0 ⇒ tanθ - 1 = 0
   ⇒ tanθ = 1
   

   ∴ tan2θ +	1	= 1 + 1 = 2
   	tan2θ

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459. What is the value of	(cotθ + cosecθ - 1)	?
     	(cotθ - cosecθ + 1)

     
     

1. 1. cotθ + cosecθ
      

   
   
   

   2. 1
      

   
   
   

   3. –1
      

   
   
   

   4. 0

   
   
   

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   Expression =	(cotθ + cosecθ - 1)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	cotθ + cosecθ - (cosec2θ - cot2θ)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	(cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	(cotθ + cosecθ)(1 - cosecθ + cotθ)	= cotθ + cosecθ
   	(cotθ - cosecθ + 1)

   Correct Option: A

   Expression =	(cotθ + cosecθ - 1)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	cotθ + cosecθ - (cosec2θ - cot2θ)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	(cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ)
   	(cotθ - cosecθ + 1)

   
   

   Expression =	(cotθ + cosecθ)(1 - cosecθ + cotθ)	= cotθ + cosecθ
   	(cotθ - cosecθ + 1)

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460. If sinθ × cosθ =	1	.The value of sinθ – cosθ is where 0° < θ < 90°
     	2

     
     

1. 1. 0
      

   
   
   

   2. √2
      

   
   
   

   3. 2
      

   
   
   

   4. 1

   
   
   

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   sinθ. cosθ =	1
   	2

   
   ⇒ 2 sinθ. cosθ = 1
   ⇒ sin2θ = 1 = sin 90°
   ⇒ 2θ = 90°
   ⇒ θ = 45°
   ∴ sinθ – cosθ = sin 45° – cos 45°
   

   sin 45° – cos 45° =	1	-	1	= 0
   	√2		√2

   Correct Option: A

   sinθ. cosθ =	1
   	2

   
   ⇒ 2 sinθ. cosθ = 1
   ⇒ sin2θ = 1 = sin 90°
   ⇒ 2θ = 90°
   ⇒ θ = 45°
   ∴ sinθ – cosθ = sin 45° – cos 45°
   

   sin 45° – cos 45° =	1	-	1	= 0
   	√2		√2

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