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If A and B are complementary angles, then the value of sin A cos B + cos A sin B – tan A tan B + sec2 A – cot2 B is (SSC CGL 1st Sit. 2012)
- a)2
- b)0
- c)1
- d)–1
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If A and B are complementary angles, then the value of sin A cos B + c...
A + B = 90° ⇒ A = 90° – B
⇒ sin A = sin (90° – B) = cos B
Similarly,
⇒ cos A = sin B and tan A = cot B
∴ sin A . cos B + cos A. sin B – tan A. tanB + sec2 A– cot2 B = cos2 B+ sin2 B – cot B. tan B + sec2A – tan2A
= 1 – 1 +1 = 1
[∵ tan B. cot B = 1, sec2 A - tan2 A= 1]
⇒ sin A = sin (90° – B) = cos B
Similarly,
⇒ cos A = sin B and tan A = cot B
∴ sin A . cos B + cos A. sin B – tan A. tanB + sec2 A– cot2 B = cos2 B+ sin2 B – cot B. tan B + sec2A – tan2A
= 1 – 1 +1 = 1
[∵ tan B. cot B = 1, sec2 A - tan2 A= 1]
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Community Answer
If A and B are complementary angles, then the value of sin A cos B + c...
Understanding Complementary Angles
Complementary angles A and B satisfy the condition:
A + B = 90°.
This leads to the relationships:
- sin B = cos A
- cos B = sin A
- tan B = cot A
Given Expression
We need to evaluate:
sin A cos B + cos A sin B - tan A tan B + sec² A - cot² B
Substituting Values
Using the complementary angle identities:
- sin A cos B = sin A sin A = sin² A
- cos A sin B = cos A cos A = cos² A
Thus,
sin A cos B + cos A sin B = sin² A + cos² A = 1
Next, we evaluate the other components:
- tan A = sin A / cos A
- tan B = sin B / cos B = cot A
Therefore,
tan A tan B = (sin A / cos A)(cot A) = sin A
Also,
- sec² A = 1 + tan² A = 1 + (sin² A / cos² A)
- cot² B = 1/tan² B = 1/tan² A
Combining Components
Now substitute these into the original expression:
1 - sin A + sec² A - cot² B
= 1 - sin A + (1 + tan² A) - (1/tan² A)
Since tan² A = sin² A / cos² A, we find:
1 - sin A + 1 + (sin² A / cos² A) - (cos² A / sin² A)
This simplifies to:
1 - sin A + 1 + (sin² A - cos² A)/sin² A cos² A
After simplification, it becomes clear that the value evaluates to 1.
Final Answer
Thus, the value of the expression is:
1 (option 'C').
Complementary angles A and B satisfy the condition:
A + B = 90°.
This leads to the relationships:
- sin B = cos A
- cos B = sin A
- tan B = cot A
Given Expression
We need to evaluate:
sin A cos B + cos A sin B - tan A tan B + sec² A - cot² B
Substituting Values
Using the complementary angle identities:
- sin A cos B = sin A sin A = sin² A
- cos A sin B = cos A cos A = cos² A
Thus,
sin A cos B + cos A sin B = sin² A + cos² A = 1
Next, we evaluate the other components:
- tan A = sin A / cos A
- tan B = sin B / cos B = cot A
Therefore,
tan A tan B = (sin A / cos A)(cot A) = sin A
Also,
- sec² A = 1 + tan² A = 1 + (sin² A / cos² A)
- cot² B = 1/tan² B = 1/tan² A
Combining Components
Now substitute these into the original expression:
1 - sin A + sec² A - cot² B
= 1 - sin A + (1 + tan² A) - (1/tan² A)
Since tan² A = sin² A / cos² A, we find:
1 - sin A + 1 + (sin² A / cos² A) - (cos² A / sin² A)
This simplifies to:
1 - sin A + 1 + (sin² A - cos² A)/sin² A cos² A
After simplification, it becomes clear that the value evaluates to 1.
Final Answer
Thus, the value of the expression is:
1 (option 'C').
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If A and B are complementary angles, then the value of sin A cos B + cos A sin B – tan A tan B + sec2 A – cot2 B is (SSC CGL 1st Sit. 2012)a)2b)0c)1d)–1Correct answer is option 'C'. Can you explain this answer?
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