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JEE MainMathematicsInverse Trigonometric Functions

The sum of all solutions to the equation ⁻¹ ( x 1 - x ) = 8 in the interval (0, 4 ) is

Options

  1. A5 2
  2. B3 4
  3. C7 2
  4. D9 2

Correct answer

C. 7 2

Step-by-step solution

Given equation: ⁻¹ ( x 1 - x ) = 8 Using the half-angle trigonometric identities, we can simplify the argument of the inverse tangent function: x = 2 ( x 2 ) ( x 2 ) 1 - x = 2 ^2 ( x 2 ) Substituting these into the fraction: x 1 - x = 2 ( x 2 ) ( x 2 ) 2 ^2 ( x 2 ) = ( x 2 ) The equation becomes: ⁻¹ ( ( x 2 ) ) = 8 Taking the tangent of both sides: ( x 2 ) = ( 8 ) Using the complementary angle identity = ( 2 - ) : ( 2 - x 2 ) = ( 8 ) The general solution for A = B is A = n + B , where n is an integer: 2 - x 2 = n +

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