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JEE MainMathematicsInverse Trigonometric Functions

Let be the real root of the equation ⁻¹(1-2x^2) + ⁻¹ ( 2x 1+x^2 ) = . Then the value of ^4 + ^2 is equal to

Options

  1. A0
  2. B2
  3. C5 -1 2
  4. D1

Correct answer

D. 1

Step-by-step solution

The given equation is ⁻¹(1-2x^2) + ⁻¹ ( 2x 1+x^2 ) = . The domain of the equation is x [-1, 1] . For x [0, 1] , we have the standard identities: ⁻¹(1-2x^2) = 2 ⁻¹x ⁻¹ ( 2x 1+x^2 ) = 2 ⁻¹x Substituting these into the equation, we get: 2 ⁻¹x + 2 ⁻¹x = ⁻¹x + ⁻¹x = 2 ⁻¹x = 2 - ⁻¹x = ⁻¹x Taking the cosine of both sides: ( ⁻¹x) = x 1 1+x^2 = x Squaring both sides, we obtain: 1 1+x^2 = x^2 x^4 + x^2 - 1 = 0 x^4 + x^2 = 1 For x [-1, 0) , ⁻¹(1-2x^2) = -2 ⁻¹x . The equation becomes -2 ⁻¹x + 2 ⁻¹x = ⁻¹x - ⁻¹x = 2 . Since x Th

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