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Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'C'. Can you explain this answer?
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Number of solutions of the equation tan x + sec x= 2 cosx lying in the...
Understanding the Equation
The given equation is:
tan x + sec x = 2 cos x
To analyze the solutions, we can rewrite the trigonometric functions in terms of sine and cosine:
- tan x = sin x / cos x
- sec x = 1 / cos x
Substituting these into the equation gives:
sin x / cos x + 1 / cos x = 2 cos x
Rearranging the Equation
Multiplying through by cos x (assuming cos x ≠ 0) leads to:
sin x + 1 = 2 cos^2 x
Using the identity cos^2 x = 1 - sin^2 x, we can rewrite the equation:
sin x + 1 = 2(1 - sin^2 x)
This simplifies to:
2 sin^2 x + sin x - 1 = 0
Finding the Roots
This quadratic equation can be solved using the quadratic formula:
- a = 2, b = 1, c = -1
The discriminant (D) is given by:
D = b^2 - 4ac = 1 + 8 = 9
Since the discriminant is positive, there are two real roots for sin x.
Checking the Interval
To find the values of x, we solve for sin x:
Using the quadratic formula, we find the roots of the equation. The roots will be in the range [-1, 1], as sin x is bounded.
Counting Solutions in the Interval [0, 2π]
For each root of sin x, there will be two corresponding angles in the interval [0, 2π] since sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
Thus, this leads to a total of two solutions from the roots found.
Conclusion
Therefore, the number of solutions to the equation tan x + sec x = 2 cos x in the interval [0, 2π] is:
Option C: 2
The given equation is:
tan x + sec x = 2 cos x
To analyze the solutions, we can rewrite the trigonometric functions in terms of sine and cosine:
- tan x = sin x / cos x
- sec x = 1 / cos x
Substituting these into the equation gives:
sin x / cos x + 1 / cos x = 2 cos x
Rearranging the Equation
Multiplying through by cos x (assuming cos x ≠ 0) leads to:
sin x + 1 = 2 cos^2 x
Using the identity cos^2 x = 1 - sin^2 x, we can rewrite the equation:
sin x + 1 = 2(1 - sin^2 x)
This simplifies to:
2 sin^2 x + sin x - 1 = 0
Finding the Roots
This quadratic equation can be solved using the quadratic formula:
- a = 2, b = 1, c = -1
The discriminant (D) is given by:
D = b^2 - 4ac = 1 + 8 = 9
Since the discriminant is positive, there are two real roots for sin x.
Checking the Interval
To find the values of x, we solve for sin x:
Using the quadratic formula, we find the roots of the equation. The roots will be in the range [-1, 1], as sin x is bounded.
Counting Solutions in the Interval [0, 2π]
For each root of sin x, there will be two corresponding angles in the interval [0, 2π] since sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
Thus, this leads to a total of two solutions from the roots found.
Conclusion
Therefore, the number of solutions to the equation tan x + sec x = 2 cos x in the interval [0, 2π] is:
Option C: 2
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Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer?
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Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer?.
Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number of solutions of the equation tan x + sec x= 2 cosx lying in the interval [0, 2π] is : (1993)a)0b)1c)2d)3Correct answer is option 'C'. Can you explain this answer?.
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