236. The numerical value of	cos² 45°	+	cos² 60°	-	tan² 30°	-		sin² 30°	is
     	sin² 60°		sin² 45°		cot² 45°			cot² 30°

1. 1. 1	1
      	4

   
   
   

   2. 	3
      	4

   
   
   

   3. 	1
      	4

   
   
   

   4. 	1
      	2

   
   
   

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1. View Hint View Answer Discuss in Forum

   		cos² 45°	+	cos²60°	-	tan²30°	-	sin²30°
   		sin²60°		sin²45°		cot²45°		cot²30°

   
   
   

   =		1	×	4	+	1	× 2 -	1	× 1 -	1
   		2		3		4		3		4 × 3

   
   

   =		2	+	1	-	1	-	1
   		3		2		3		12

   
   

   =		8 + 6 - 4 - 1	=	9	=	3
   		12		12		4

   
   

   Correct Option: B

   		cos² 45°	+	cos²60°	-	tan²30°	-	sin²30°
   		sin²60°		sin²45°		cot²45°		cot²30°

   
   
   

   =		1	×	4	+	1	× 2 -	1	× 1 -	1
   		2		3		4		3		4 × 3

   
   

   =		2	+	1	-	1	-	1
   		3		2		3		12

   
   

   =		8 + 6 - 4 - 1	=	9	=	3
   		12		12		4

   
   

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237. The value of tan1°tan2°tan3° ........tan89° is

1. 1. 1

   
   
   

   2. - 1

   
   
   

   3. 0

   
   
   

   4. None of the options

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan (90° – θ) = cotθ tanθ.cotθ = 1
   tan 89° = tan (90° – 1°) = cot 1°.
   tan 88° = tan (90° – 2°) = cot 2°.
   ∴ Expression = (tan 1°.tan 89°) (tan 2°.tan 88°) ---- tan 45°
   = (tan 1°.cot 1°). (tan 2°.cot 2°) ---- tan 45°
   = 1.1 ----1 = 1

   Correct Option: A

   tan (90° – θ) = cotθ tanθ.cotθ = 1
   tan 89° = tan (90° – 1°) = cot 1°.
   tan 88° = tan (90° – 2°) = cot 2°.
   ∴ Expression = (tan 1°.tan 89°) (tan 2°.tan 88°) ---- tan 45°
   = (tan 1°.cot 1°). (tan 2°.cot 2°) ---- tan 45°
   = 1.1 ----1 = 1

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238. If		cos α	= n and	cos α	= m , then the vlaue of cos² β is
     		sin β		cos β

1. 1. 	m²
      	m² + n²

   
   
   

   2. 	1
      	m² + n²

   
   
   

   3. 	n²
      	m² + n²

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   		cos α	= n and	cos α	= m
   		sin β		sin β

   
   ⇒ cosα = n sinβ and cosα = m cosβ.
   ∴ n² sin² β = m² cos² β
   ⇒ n² (1 – cos² β) = m² cos² β
   ⇒ n² – n² cos² β = m² cos² β ⇒ m² cos² β + n² cos² β = n²
   ⇒ cos² β (m² + n² ) = n²
   

   ⇒ cos²β =	n²
   	m² + n²

   
   

   Correct Option: C

   		cos α	= n and	cos α	= m
   		sin β		sin β

   
   ⇒ cosα = n sinβ and cosα = m cosβ.
   ∴ n² sin² β = m² cos² β
   ⇒ n² (1 – cos² β) = m² cos² β
   ⇒ n² – n² cos² β = m² cos² β ⇒ m² cos² β + n² cos² β = n²
   ⇒ cos² β (m² + n² ) = n²
   

   ⇒ cos²β =	n²
   	m² + n²

   
   

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239. If 0° ≤ A ≤ 90°, the simplified form of the given expression sin A cos A (tan A – cot A) is

1. 1. 1

   
   
   

   2. 1 – 2 sin² A

   
   
   

   3. 2 sin² A – 1

   
   
   

   4. 1 – cos² A

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin A. cos A (tan A – cot A)
   

   = sin A. cos A		sin A	-	cos A
   		cos A		sin A

   
   

   = sin A. cos A			sin² A - cos² A
   			sin A . cos A

   
   = sin²A – cos²A
   = sin²A – (1 – sin²A)
   = sin²A – 1 + sin²A
   = 2 sin²A – 1

   Correct Option: C

   sin A. cos A (tan A – cot A)
   

   = sin A. cos A		sin A	-	cos A
   		cos A		sin A

   
   

   = sin A. cos A			sin² A - cos² A
   			sin A . cos A

   
   = sin²A – cos²A
   = sin²A – (1 – sin²A)
   = sin²A – 1 + sin²A
   = 2 sin²A – 1

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240. If θ is an acute angle and tan²θ + (1 / tan² θ) = 2, then the value of θ is :

1. 1. 60°

   
   
   

   2. 45°

   
   
   

   3. 15°

   
   
   

   4. 30°

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan² θ +	1	= 2
   	tan² θ

   
   

   ⇒	tan4 θ + 1	= 2
   	tan² θ

   
   ⇒ tan⁴ θ + 1 = 2 tan² θ
   ⇒ tan⁴ θ – 2 tan² θ + 1 = 0
   ⇒ (tan² θ – 1)2 = 0
   ⇒ tan² θ – 1 = 0
   ⇒ tan² θ = 1
   ⇒ tanθ = 1 = tan 45°
   ⇒ ² θ = 45°
   ∵ θ is an acute angle

   Correct Option: B

   tan² θ +	1	= 2
   	tan² θ

   
   

   ⇒	tan4 θ + 1	= 2
   	tan² θ

   
   ⇒ tan⁴ θ + 1 = 2 tan² θ
   ⇒ tan⁴ θ – 2 tan² θ + 1 = 0
   ⇒ (tan² θ – 1)2 = 0
   ⇒ tan² θ – 1 = 0
   ⇒ tan² θ = 1
   ⇒ tanθ = 1 = tan 45°
   ⇒ ² θ = 45°
   ∵ θ is an acute angle

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