506. If sinA + sin²A = 1, then what is the value of cos²A + cos⁴A ?

1. 1. 1

   
   
   

   2. 2

   
   
   

   3. 1

      2

   
   
   

   4. 1

      4

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinA + sin²A
   ⇒  sinA = 1 – sin²A = cos²A
   ∴  cos²A + cos⁴A
   = cos²A + (cos²A)²
   = cos²A + sin²A

   Correct Option: A

   sinA + sin²A
   ⇒  sinA = 1 – sin²A = cos²A
   ∴  cos²A + cos⁴A
   = cos²A + (cos²A)²
   = cos²A + sin²A

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507. If sinA + sin²A = 1, then what is the value of cos²A + cos⁴A ?

1. 1. 1

   
   
   

   2. 2

   
   
   

   3. 1

      2

   
   
   

   4. 1

      4

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinA + sin²A
   ⇒  sinA = 1 – sin²A = cos²A
   ∴  cos²A + cos⁴A
   = cos²A + (cos²A)²
   = cos²A + sin²A

   Correct Option: A

   sinA + sin²A
   ⇒  sinA = 1 – sin²A = cos²A
   ∴  cos²A + cos⁴A
   = cos²A + (cos²A)²
   = cos²A + sin²A

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508. If sinC – sinD = x, then the value of x is
     

1. 1. 2sin		(C + D)		cos		(C − D)
      		2				2

   
   
   

   2. 2cos		(C + D)		cos		(C − D)
      		2				2

   
   
   

   3. 2cos		(C + D)		sin		(C − D)
      		2				2

   
   
   

   4. 2sin		(C + D)		sin		(D − C)
      		2				2

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinC – sinD = 2 cos	C + D	. sin	C – D	= x
   	2		2

   
   Illustration
   sin (A + B)
   = sinA . cosB + cosA . sinB
   sin (A – B)
   = sinA . cosB – cosA . sinB
   ∴  sin (A + B) – sin (A – B)
   = 2 cosA . sinB
   Let, A + B = C ; A – B = D
   ∴  On adding,
   

   A =	C + D
   	2

   
   On subtracting,
   

   B =	C – D
   	2

   
   

   ∴  sinC – sinD = 2 cos	C + D	.	C − D
   	2		2

   Correct Option: C

   sinC – sinD = 2 cos	C + D	. sin	C – D	= x
   	2		2

   
   Illustration
   sin (A + B)
   = sinA . cosB + cosA . sinB
   sin (A – B)
   = sinA . cosB – cosA . sinB
   ∴  sin (A + B) – sin (A – B)
   = 2 cosA . sinB
   Let, A + B = C ; A – B = D
   ∴  On adding,
   

   A =	C + D
   	2

   
   On subtracting,
   

   B =	C – D
   	2

   
   

   ∴  sinC – sinD = 2 cos	C + D	.	C − D
   	2		2

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509. If	sinθ + cosθ	= 3 then the value of sin4θ − cos4θ is
     	sinθ − cosθ

1. 1. 4

      3

   
   
   

   2. 3

      4

      
      

   
   
   

   3. 5

      3

   
   
   

   4. 3

      5

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinθ + cosθ	= 3
   sinθ − cosθ

   
   ⇒  3 sinθ – 3 cosθ = sinθ + cosθ
   ⇒  3 sinθ – sinθ = 3 cosθ + cosθ
   ⇒  2 sinθ = 4 cosθ
   ⇒  sinθ = 2 cosθ
   ⇒  tanθ = 2
   ∴  sec²θ = 1 + tan²θ
   = 1 + 4 = 5
   

   ∴  cos2θ =	1
   	5

   
   ∴  sin²θ = 1 – cos²θ
   

   = 1 −	1
   	5

   
   

   =	4
   	5

   
   ∴  sin⁴θ − cos⁴θ
   = (sin²θ + cos²θ)(sin²θ – cos²θ)
   = sin²θ – cos²θ
   

   =	4	−	1	=	3
   	5		5		5

   Correct Option: D

   sinθ + cosθ	= 3
   sinθ − cosθ

   
   ⇒  3 sinθ – 3 cosθ = sinθ + cosθ
   ⇒  3 sinθ – sinθ = 3 cosθ + cosθ
   ⇒  2 sinθ = 4 cosθ
   ⇒  sinθ = 2 cosθ
   ⇒  tanθ = 2
   ∴  sec²θ = 1 + tan²θ
   = 1 + 4 = 5
   

   ∴  cos2θ =	1
   	5

   
   ∴  sin²θ = 1 – cos²θ
   

   = 1 −	1
   	5

   
   

   =	4
   	5

   
   ∴  sin⁴θ − cos⁴θ
   = (sin²θ + cos²θ)(sin²θ – cos²θ)
   = sin²θ – cos²θ
   

   =	4	−	1	=	3
   	5		5		5

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510. If sinθ + cosθ = 1, then sinθ. cosθ. is equal to

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 1

      2

   
   
   

   4. −	1
      	2

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinθ + cosθ = 1
   On squaring,
   (sinθ + cosθ) = 1
   ⇒  sin²θ + cos²θ + 2 sinθ. cosθ = 1
   ⇒  1 + 2 sinθ . cosθ = 1
   ⇒  2 sinθ . cosθ = 1 – 1 = 0
   ⇒  sinθ. cosθ = 0

   Correct Option: A

   sinθ + cosθ = 1
   On squaring,
   (sinθ + cosθ) = 1
   ⇒  sin²θ + cos²θ + 2 sinθ. cosθ = 1
   ⇒  1 + 2 sinθ . cosθ = 1
   ⇒  2 sinθ . cosθ = 1 – 1 = 0
   ⇒  sinθ. cosθ = 0

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