136. If sin (60° – θ) = cos (&psy; – 30°), then the value of tan (&psy; – θ) is (assume that θ and &psy; are both positive acute angles with θ < 60° and &psy; > 30°).

1. 1. 	1
      	√3

      
      

   
   
   

   2. 0

   
   
   

   3. √3

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin(60° - θ ) = cos (ψ - 30°)
   = sin (90° - &pasi; + 30°)
   = sin (120° - ψ)
   ⇒ 60° - θ = 120° - ψ
   ⇒ ψ - θ = 60°
   ∴ tan (ψ - θ) = tan 60° = √3
   

   Correct Option: C

   sin(60° - θ ) = cos (ψ - 30°)
   = sin (90° - &pasi; + 30°)
   = sin (120° - ψ)
   ⇒ 60° - θ = 120° - ψ
   ⇒ ψ - θ = 60°
   ∴ tan (ψ - θ) = tan 60° = √3
   

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137. If sec θ + tan θ = &radic3 (0° ≤ θ ≤ 90°), then tan 3θ is

1. 1. undefined

   
   
   

   2. 	1
      	√3

   
   
   

   3. 	1
      	√2

      
      

   
   
   

   4. 3

   
   
   

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1. View Hint View Answer Discuss in Forum

   sec θ + tan θ = √3.....(i)
   ∵ sec² θ - tan² θ = 1
   ⇒ (secθ - tanθ )(secθ + tanθ ) = 1
   

   ⇒ secθ - tanθ =	1	....(ii)
   	√3

   
   By subtracting (ii) from (i)
   secθ + tanθ - secθ + tanθ
   

   = √3 -	1
   	√3

   
   

   ⇒ 2tan θ =	3 - 1
   	√3

   
   

   = tan θ =	1	= tan 30°
   	√3

   
   ⇒ θ = 30°
   ∴ tan3 θ =tan 90° = undefined

   Correct Option: A

   sec θ + tan θ = √3.....(i)
   ∵ sec² θ - tan² θ = 1
   ⇒ (secθ - tanθ )(secθ + tanθ ) = 1
   

   ⇒ secθ - tanθ =	1	....(ii)
   	√3

   
   By subtracting (ii) from (i)
   secθ + tanθ - secθ + tanθ
   

   = √3 -	1
   	√3

   
   

   ⇒ 2tan θ =	3 - 1
   	√3

   
   

   = tan θ =	1	= tan 30°
   	√3

   
   ⇒ θ = 30°
   ∴ tan3 θ =tan 90° = undefined

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138. The value of (sin² 25° + sin² 65°) is :

1. 1. 	√3
      	2

   
   
   

   2. 1

   
   
   

   3. 0

   
   
   

   4. 	2
      	√3

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin² 25° + sin² 65°
   = sin² 25° + sin² (90° - 25°)
   = sin² 25° + cos² 25° = 1

   Correct Option: B

   sin² 25° + sin² 65°
   = sin² 25° + sin² (90° - 25°)
   = sin² 25° + cos² 25° = 1

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139. The value of cos 1° cos 2° cos 3°....... cos 177° cos 178° cos 179° is :

1. 1. 0

   
   
   

   2. 	1
      	2

   
   
   

   3. 1

   
   
   

   4. 	1
      	√2

   
   
   

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1. View Hint View Answer Discuss in Forum

   cos 90° = 0
   ∴ cos 1°. cos 2° ... cos 179° = 0

   Correct Option: A

   cos 90° = 0
   ∴ cos 1°. cos 2° ... cos 179° = 0

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140. If sec θ = x +	1	(0° < θ < 90°), then sec θ + tan θ is equal to
     	4x

     
     

1. 1. 	x
      	2

   
   
   

   2. 2x

   
   
   

   3. x

   
   
   

   4. 	1
      	2x

   
   
   

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1. View Hint View Answer Discuss in Forum

   sec θ =	4x² + 1
   	4x

   
   tan θ = √sec² θ - 1
   √[(4x² + 1 / 4x)²] - 1
   √[(4x² + 1)² - (4x)² / (4x)²]
   

   		(2x + 1)(2x - 1)	=	4x² - 1
   		4x		4x

   
   

   ∴ sec θ + tan θ =		4x² + 1	+	4x² - 1
   		4x		4x

   
   

   =	4x² + 1 + 4x² - 1
   	4x

   
   

   =	8x²	= 2x
   	4x

   
   

   Correct Option: B

   sec θ =	4x² + 1
   	4x

   
   tan θ = √sec² θ - 1
   √[(4x² + 1 / 4x)²] - 1
   √[(4x² + 1)² - (4x)² / (4x)²]
   

   		(2x + 1)(2x - 1)	=	4x² - 1
   		4x		4x

   
   

   ∴ sec θ + tan θ =		4x² + 1	+	4x² - 1
   		4x		4x

   
   

   =	4x² + 1 + 4x² - 1
   	4x

   
   

   =	8x²	= 2x
   	4x

   
   

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