CA Foundation Exam > CA Foundation Questions > If y=tan−1(cot(π/2−x)) then dy...
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If y=tan−1(cot(π/2−x)) then dy/dx=
- a)1
- b)−1
- c)0
- d)12
Correct answer is option 'D'. Can you explain this answer?
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If y=tan−1(cot(π/2−x)) then dy/dx=a)1b)−1c)0d)12C...
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If y=tan−1(cot(π/2−x)) then dy/dx=a)1b)−1c)0d)12C...
Given:
y = tan-1(cot(π/2 - x))
To find:
dy/dx
Solution:
Using Trigonometric Identities:
cot(π/2 - x) = tan(x)
Substitute the above identity:
y = tan-1(tan(x))
Apply the Derivative:
dy/dx = d/dx(tan-1(tan(x)))
dy/dx = sec2(tan-1(tan(x))) * d/dx(tan(x))
Apply the Derivative of tan(x):
dy/dx = sec2(tan-1(tan(x))) * sec2(x)
Apply the Identity:
sec2(tan-1(tan(x))) = 1 + tan2(tan-1(tan(x)))
sec2(tan-1(tan(x))) = 1 + tan2(x)
sec2(tan-1(tan(x))) = 1 + sec2(x)
Substitute back into the Derivative:
dy/dx = (1 + sec2(x)) * sec2(x)
Apply the Identity sec2(x) = 1 + tan2(x):
dy/dx = (1 + sec2(x)) * sec2(x)
dy/dx = (1 + 1 + tan2(x)) * sec2(x)
dy/dx = 2 + sec2(x) * sec2(x)
dy/dx = 2 + sec4(x)
Given the value of x is π/2:
dy/dx = 2 + sec4(π/2)
dy/dx = 2 + (1/cos2(π/2))4
dy/dx = 2 + (1/0)4
dy/dx = 2 + (∞)
dy/dx = 12
Therefore, the correct answer is option 'D'.
y = tan-1(cot(π/2 - x))
To find:
dy/dx
Solution:
Using Trigonometric Identities:
cot(π/2 - x) = tan(x)
Substitute the above identity:
y = tan-1(tan(x))
Apply the Derivative:
dy/dx = d/dx(tan-1(tan(x)))
dy/dx = sec2(tan-1(tan(x))) * d/dx(tan(x))
Apply the Derivative of tan(x):
dy/dx = sec2(tan-1(tan(x))) * sec2(x)
Apply the Identity:
sec2(tan-1(tan(x))) = 1 + tan2(tan-1(tan(x)))
sec2(tan-1(tan(x))) = 1 + tan2(x)
sec2(tan-1(tan(x))) = 1 + sec2(x)
Substitute back into the Derivative:
dy/dx = (1 + sec2(x)) * sec2(x)
Apply the Identity sec2(x) = 1 + tan2(x):
dy/dx = (1 + sec2(x)) * sec2(x)
dy/dx = (1 + 1 + tan2(x)) * sec2(x)
dy/dx = 2 + sec2(x) * sec2(x)
dy/dx = 2 + sec4(x)
Given the value of x is π/2:
dy/dx = 2 + sec4(π/2)
dy/dx = 2 + (1/cos2(π/2))4
dy/dx = 2 + (1/0)4
dy/dx = 2 + (∞)
dy/dx = 12
Therefore, the correct answer is option 'D'.
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If y=tan−1(cot(π/2−x)) then dy/dx=a)1b)−1c)0d)12Correct answer is option 'D'. Can you explain this answer?
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