476. If sinθ + cosecθ = 2, the value of sin¹⁰⁰θ + cosec¹⁰⁰θ is :
     

1. 1. 1
      

   
   
   

   2. 2
      

   
   
   

   3. 3
      

   
   
   

   4. 4

   
   
   

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   sinθ + cosecθ = 2
   

   ⇒ sinθ +	1	= 2
   	sinθ

   
   ⇒ sin²θ + 1 = 2sinθ
   ⇒ sin²θ - 2sinθ + 1 = 0
   ⇒ (sinθ - 1)² = 0
   ⇒ sinθ = 1
   

   ∴ cosecθ =	1	= 1
   	sinθ

   
   ∴ sin¹⁰⁰θ + cosec¹⁰⁰θ = 1 + 1 = 2

   Correct Option: B

   sinθ + cosecθ = 2
   

   ⇒ sinθ +	1	= 2
   	sinθ

   
   ⇒ sin²θ + 1 = 2sinθ
   ⇒ sin²θ - 2sinθ + 1 = 0
   ⇒ (sinθ - 1)² = 0
   ⇒ sinθ = 1
   

   ∴ cosecθ =	1	= 1
   	sinθ

   
   ∴ sin¹⁰⁰θ + cosec¹⁰⁰θ = 1 + 1 = 2

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477. The value of	sin 65°	is
     	cos 25°

     
     

1. 1. 0
      

   
   
   

   2. 1
      

   
   
   

   3. 2
      

   
   
   

   4. Not defined

   
   
   

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1. View Hint View Answer Discuss in Forum

   	sin 65°	=	sin( 90° - 25° )
   	cos 25°		cos 25°

   
   

   ⇒	sin 65°	=	cos25°	= 1
   	cos 25°		cos 25°

   Correct Option: B

   	sin 65°	=	sin( 90° - 25° )
   	cos 25°		cos 25°

   
   

   ⇒	sin 65°	=	cos25°	= 1
   	cos 25°		cos 25°

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478. If	sinx	+	sinx	= 4 ; and 0° < x < 90°, then find the value of x
     	1 + cosx		1 - cosx

     
     

1. 1. 10°
      

   
   
   

   2. 15°
      

   
   
   

   3. 45°
      

   
   
   

   4. 30°

   
   
   

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1. View Hint View Answer Discuss in Forum

   	sinx	+	sinx	= 4
   	1 + cosx		1 - cosx

   
   

   ⇒	sinx(1 - cosx) + sinx(1 + cosx)	= 4
   	(1 + cosx)(1 - cosx)

   
   

   ⇒	sinx - cosx.sinx + sinx + cosx.sinx	= 4
   	(1 - cos2x)

   
   

   ⇒	2sinx	= 4
   	sin2x

   
   ⇒ 2sinx = 1
   

   ⇒ sinx =	1	= sin30° ⇒ x = 30°
   	2

   Correct Option: D

   	sinx	+	sinx	= 4
   	1 + cosx		1 - cosx

   
   

   ⇒	sinx(1 - cosx) + sinx(1 + cosx)	= 4
   	(1 + cosx)(1 - cosx)

   
   

   ⇒	sinx - cosx.sinx + sinx + cosx.sinx	= 4
   	(1 - cos2x)

   
   

   ⇒	2sinx	= 4
   	sin2x

   
   ⇒ 2sinx = 1
   

   ⇒ sinx =	1	= sin30° ⇒ x = 30°
   	2

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479. Find the value of tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ).
     

1. 1. tan 10θ
      

   
   
   

   2. tan8θ
      

   
   
   

   3. tan 12θ
      

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ

   1 +

   1

   1 +

   1

   1 +

   1

   cos2θ

   cos4θ

   cos8θ

   
   

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ		cos2θ + 1			cos4θ + 1					cos8θ + 1
   		cos2θ			cos4θ					cos8θ

   
   

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ		2cos2θ			2cos22θ					2cos24θ
   		cos2θ			cos4θ					cos8θ

   
   [ ∴ 1 + cos2θ = 2cos²θ ]
   

   Required answer = 8 .	tanθ.cos2θ .cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 4 .	2sinθcosθ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 4 .	sin2θ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 2 .	2sin2θ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer =	2sin4θ.cos4θ
   	cos8θ

   
   

   Required answer =	sin8θ	= tan8θ
   	cos8θ

   Correct Option: B

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ

   1 +

   1

   1 +

   1

   1 +

   1

   cos2θ

   cos4θ

   cos8θ

   
   

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ		cos2θ + 1			cos4θ + 1					cos8θ + 1
   		cos2θ			cos4θ					cos8θ

   
   

   tan θ(1 + sec2θ)(1 + sec 4θ) (1 + sec8θ) = tanθ		2cos2θ			2cos22θ					2cos24θ
   		cos2θ			cos4θ					cos8θ

   
   [ ∴ 1 + cos2θ = 2cos²θ ]
   

   Required answer = 8 .	tanθ.cos2θ .cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 4 .	2sinθcosθ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 4 .	sin2θ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer = 2 .	2sin2θ.cos2θ.cos4θ
   	cos8θ

   
   

   Required answer =	2sin4θ.cos4θ
   	cos8θ

   
   

   Required answer =	sin8θ	= tan8θ
   	cos8θ

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480. If tanθ1 = 1, sinθ2 =	1	, then the value of sin (θ1 + θ2) equal to
     	√2

     
     

1. 1. –1
      

   
   
   

   2. 0
      

   
   
   

   3. 1
      

   
   
   

   4. 1

      2

   
   
   

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1. View Hint View Answer Discuss in Forum

   tanθ₁ = 1 = tan 45°
   ⇒ θ₁ = 45°
   

   Again , sinθ2 =	1	= sin45°
   	√2

   
   ⇒ θ₂ = 45°
   ∴ sin(θ₁ + θ₂) = sin90° = 1

   Correct Option: C

   tanθ₁ = 1 = tan 45°
   ⇒ θ₁ = 45°
   

   Again , sinθ2 =	1	= sin45°
   	√2

   
   ⇒ θ₂ = 45°
   ∴ sin(θ₁ + θ₂) = sin90° = 1

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