Solve que 37:

37. The solution set of the equation  sin - 1 1 - x 2 + cos - 1 x = cos - 1   1 - x 2 x - sin - 1 x
(A) [-1,1]-{0}                  (B) (0,1]  {-1}          (C) [-1,0)  {1}        (D) ​[-1,1]

Dear Student,
Please find below the solution to the asked query:

We havesin-11-x2+cos-1x=cot-1 1-x2x-sin-1xSet cos-1x=A As range of cos-1x  is 0,πHence A0,πx=cosA, equation becomes sin-11-cos2A+cos-1cosA=cot-1 1-cos2AcosA-sin-1cosAsin-1sin2A+A=cot-1 sin2AcosA-π2-cos-1cosAAs sin-1α+cos-1α=π2 and cos-1cosα=α when α0,πsin-1sinA+A=cot-1 sinAcosA-π2-AAs sinA will be non-negative when A0,π, hence sinA=sinAsin-1sinA+A=cot-1 sinAcosA-π2+Asin-1sinA=cot-1 tanA-π2sin-1sinA=π2-tan-1 tanA-π2As cot-1α+tan-1α=π2sin-1sinA=-tan-1 tanACase 1:When A[0,π2)A=-A\As sin-1sinA=A and tan-1tanA=A when A[0,π2)2A=0A=0x=cosA=cos0=1Case 2:When Aπ2,ππ-A=-A-π  As sin-1sinA=π-A and  tan-1tanA=A-π when  Aπ2,ππ-A=π-A which is trueHence  Aπ2,πx=cosAcosπ,cosπ2x-1,0But x=0 will be taken because cot-1 1-x2x is not definedHence x[-1,0)Combining two cases, we get  x[-1,0)1

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