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If 2tan−1(cos x) = tan−1(2cosecx) , then x =
- a)
- b)
- c)
- d)none of these.
Correct answer is option 'B'. Can you explain this answer?
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If2tan−1(cos x) = tan−1(2cosecx), then x =a)b)c)d)none of ...
If 2 tan-1 (cos x) = tan -1(2 cosec x),
2tan-1(cos x) = tan-1 (2 cosec x)
= tan-1(2 cosec x)
= cot x cosec x = cosec x = x = π/4
Most Upvoted Answer
If2tan−1(cos x) = tan−1(2cosecx), then x =a)b)c)d)none of ...
Given Equation:
2tan-1(cos x) = tan-1(2cosec x)
Solution:
Using Trigonometric Identities:
We know that tan-1(x) + tan-1(y) = tan-1((x + y) / (1 - xy))
Applying the Identity:
tan-1(cos x) + tan-1(2cosec x) = tan-1((cos x + 2cosec x) / (1 - cos x * 2cosec x))
Further Simplification:
tan-1(cos x) + tan-1(2cosec x) = tan-1((cos x + 2 / sin x) / (1 - 2cos x / sin x))
tan-1(cos x) + tan-1(2cosec x) = tan-1((sin x * cos x + 2) / (sin x - 2cos x))
Given Equation Becomes:
tan-1((sin x * cos x + 2) / (sin x - 2cos x)) = 2tan-1(cos x)
Comparing the Arguments:
(sin x * cos x + 2) / (sin x - 2cos x) = 2cos x
sin x * cos x + 2 = 2cos x * (sin x - 2cos x)
sin x * cos x + 2 = 2sin x cos x - 4cos2x
Further Simplification:
sin x cos x + 4cos2x - 2sin x cos x - 2 = 0
2cos2x - 2sin x cos x - 2 = 0
cos2x - sin x cos x - 1 = 0
cos2x - 1 = 0
cos2x = 1
cos x = ±1
Since cos x cannot be equal to -1:
cos x = 1
Therefore, x = π/4:
Hence, the correct answer is option 'b'.
2tan-1(cos x) = tan-1(2cosec x)
Solution:
Using Trigonometric Identities:
We know that tan-1(x) + tan-1(y) = tan-1((x + y) / (1 - xy))
Applying the Identity:
tan-1(cos x) + tan-1(2cosec x) = tan-1((cos x + 2cosec x) / (1 - cos x * 2cosec x))
Further Simplification:
tan-1(cos x) + tan-1(2cosec x) = tan-1((cos x + 2 / sin x) / (1 - 2cos x / sin x))
tan-1(cos x) + tan-1(2cosec x) = tan-1((sin x * cos x + 2) / (sin x - 2cos x))
Given Equation Becomes:
tan-1((sin x * cos x + 2) / (sin x - 2cos x)) = 2tan-1(cos x)
Comparing the Arguments:
(sin x * cos x + 2) / (sin x - 2cos x) = 2cos x
sin x * cos x + 2 = 2cos x * (sin x - 2cos x)
sin x * cos x + 2 = 2sin x cos x - 4cos2x
Further Simplification:
sin x cos x + 4cos2x - 2sin x cos x - 2 = 0
2cos2x - 2sin x cos x - 2 = 0
cos2x - sin x cos x - 1 = 0
cos2x - 1 = 0
cos2x = 1
cos x = ±1
Since cos x cannot be equal to -1:
cos x = 1
Therefore, x = π/4:
Hence, the correct answer is option 'b'.
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If2tan−1(cos x) = tan−1(2cosecx), then x =a)b)c)d)none of these.Correct answer is option 'B'. Can you explain this answer?
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