Q34 / 90 · JEE Main 2021 (27 Aug Shift 2) · Mathematics

If y(x)=cot-11+sinx+1-sinx1+sinx-1-sinx, x∈π2,π, then dydx at x=5π6 is:

  • A
    0
  • B
    -1
  • Correct answer
    C
    -12
  • D
    12

Solution

y(x)=cot−1⁡1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x

As we know that, 1-sin⁡x =sinx2- cosx2 & 1+sin⁡x =sinx2+ cosx2

y(x)=cot−1⁡cos⁡x2+sin⁡x2+cos⁡x2−sin⁡x2cos⁡x2+sin⁡x2−cos⁡x2−sin⁡x2

=cot−1⁡cos⁡x2+sin⁡x2+sin⁡x2−cos⁡x2cos⁡x2+sin⁡x2−sin⁡x2+cos⁡x2 for x∈π2,π

=cot−1⁡sin⁡x2cos⁡x2

=cot−1⁡tan⁡x2

=cot−1⁡cot⁡(π2-x2)

y(x)=π2−x2

y′(x)=−12

Hence, the correct answer is option: 'C'.

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