76. The value of (cos 53° – sin 37°) is

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 2 sin 37°

   
   
   

   4. 2 cos 53°

   
   
   

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   cos53° – sin37°
   = cos (90° – 37°) – sin37°
   = sin37° – sin37° = 0

   Correct Option: A

   cos53° – sin37°
   = cos (90° – 37°) – sin37°
   = sin37° – sin37° = 0

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77. If cosecθ + sinθ = 5/2, then the value of (cosecθ – sinθ) is :

1. 1. -	3
      	2

   
   
   

   2. 3

      2

   
   
   

   3. -	√3
      	2

   
   
   

   4. √3

      2

   
   
   

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1. View Hint View Answer Discuss in Forum

   cosecθ + sinθ =	5
   	2

   
   

   ⇒	1	+ sinθ =	5
   	2		sinθ

   
   

   ⇒	1 + sinθ	=	5
   	2		sinθ

   
   ⇒ 2 sin²θ + 2 = 5sinθ
   ⇒ 2 sin2θ – 5 sinθ + 2 = 0
   ⇒ 2 sin2θ – 4 sinθ – sinθ + 2 = 0
   ⇒ 2 sinθ (sinθ – 2) – 1 (sinθ – 2) = 0
   ⇒ (2 sinθ – 1) (sinθ – 2) = 0
   ⇒ 2 sinθ – 1 = 0
   ⇒ 2 sinθ = 1
   

   ⇒ sinθ =	1	because sinθ ≠ 2
   	2

   
   ⇒ cosecθ = 2
   

   ∴ cosecθ – sinθ = 2 –	1	=	3
   	2		2

   Correct Option: B

   cosecθ + sinθ =	5
   	2

   
   

   ⇒	1	+ sinθ =	5
   	2		sinθ

   
   

   ⇒	1 + sinθ	=	5
   	2		sinθ

   
   ⇒ 2 sin²θ + 2 = 5sinθ
   ⇒ 2 sin2θ – 5 sinθ + 2 = 0
   ⇒ 2 sin2θ – 4 sinθ – sinθ + 2 = 0
   ⇒ 2 sinθ (sinθ – 2) – 1 (sinθ – 2) = 0
   ⇒ (2 sinθ – 1) (sinθ – 2) = 0
   ⇒ 2 sinθ – 1 = 0
   ⇒ 2 sinθ = 1
   

   ⇒ sinθ =	1	because sinθ ≠ 2
   	2

   
   ⇒ cosecθ = 2
   

   ∴ cosecθ – sinθ = 2 –	1	=	3
   	2		2

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78. The value of	2tan 53°	-	cot 80°	is :
    	cot37 °		tan 10 °

1. 1. 3

   
   
   

   2. 2

   
   
   

   3. 1

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   2tan 53°	-	cot 80°
   cot 37°		tan 10°

   
   

   =	2tan (90° - 37°)	-	cot (90° - 10°)
   	cot 37°		tan 10°

   
   

   =	2cot 37°	-	tan 10°
   	cot 37°		tan 10°

   
   = 2 – 1 = 1

   Correct Option: C

   2tan 53°	-	cot 80°
   cot 37°		tan 10°

   
   

   =	2tan (90° - 37°)	-	cot (90° - 10°)
   	cot 37°		tan 10°

   
   

   =	2cot 37°	-	tan 10°
   	cot 37°		tan 10°

   
   = 2 – 1 = 1

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79. The value of cot 10°. cot 20°. cot 60°. cot 70°. cot 80° is :

1. 1. 1

   
   
   

   2. –1

   
   
   

   3. √3

   
   
   

   4. 1

      √3

   
   
   

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1. View Hint View Answer Discuss in Forum

   Expression = cot10°.cot20°.cot60°.cot70°.cot80° = (cot10°.cot80°) (cot20°. cot70°).cot60°
   

   = {cot10°.cot (90°–10°)} {cot20°.cot(90° – 20°)}.	1	= (cot10°.tan10°)(cot20°.tan20°).	1
   	√3		√3

   
   

   = 1.1	1	=	1
   	√3		√3

   
   [∵ cot (90° – θ) = tanθ; tanθ.cotθ = 1]

   Correct Option: D

   Expression = cot10°.cot20°.cot60°.cot70°.cot80° = (cot10°.cot80°) (cot20°. cot70°).cot60°
   

   = {cot10°.cot (90°–10°)} {cot20°.cot(90° – 20°)}.	1	= (cot10°.tan10°)(cot20°.tan20°).	1
   	√3		√3

   
   

   = 1.1	1	=	1
   	√3		√3

   
   [∵ cot (90° – θ) = tanθ; tanθ.cotθ = 1]

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80. If 7sin²θ + 3cos²θ = 4, and 0° < θ < 90°, then the value of tanθ is :

1. 1. 1

      √2

   
   
   

   2. 1

      √3

   
   
   

   3. √	3
      	2

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   7 sin²θ + 3 cos²θ = 4
   On dividing by cos²θ,
   

   7 sin²θ	+	3 cos²θ	=	4
   cos²θ		cos²θ		cos²θ

   
   ⇒ 7tan²θ + 3 = 4 sec²θ = 4 (1 + tan²θ)
   ⇒ 7tan²θ + 3 = 4 + 4 tan2θ
   ⇒ 7tan²θ – 4 tan2θ = 4 – 3
   ⇒ 3tan²θ = 1
   

   ⇒ tan²θ =	1
   	3

   
   

   ⇒ tan²θ =	1
   	√3

   Correct Option: B

   7 sin²θ + 3 cos²θ = 4
   On dividing by cos²θ,
   

   7 sin²θ	+	3 cos²θ	=	4
   cos²θ		cos²θ		cos²θ

   
   ⇒ 7tan²θ + 3 = 4 sec²θ = 4 (1 + tan²θ)
   ⇒ 7tan²θ + 3 = 4 + 4 tan2θ
   ⇒ 7tan²θ – 4 tan2θ = 4 – 3
   ⇒ 3tan²θ = 1
   

   ⇒ tan²θ =	1
   	3

   
   

   ⇒ tan²θ =	1
   	√3

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