Q105 / 160 · AP EAMCET 2021 (24 Aug Shift 2) · Mathematics

If $y=\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}$, then the values of $\frac{d y}{d x}$ and $\frac{d^2 y}{d x^2}$ respectively are

  • A
    1,0
  • B
    $\frac{x}{2}, \frac{1}{2}$
  • Correct answer
    C
    $\frac{1}{2}, 0$
  • D
    $\frac{-1}{2}, 0$

Solution

We have $y=\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}$ $\Rightarrow \quad y=\tan ^{-1} \sqrt{\frac{2 \sin ^2 x / 2}{2 \cos ^2 x / 2}}=\tan ^{-1}(\tan x / 2)$ $\Rightarrow \quad y=x / 2 \Rightarrow \frac{d y}{d x}=\frac{1}{2}$ $\Rightarrow \quad \frac{d^2 y}{d x^2}=0$

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

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