Q104 / 160 · AP EAMCET 2024 (21 May Shift 1) · Mathematics

Which one of the following is false?

  • A
    $\frac{d}{d x}\left[\sec ^{-1}(\cos h x)\right]=\sec h x$
  • B
    $\frac{d}{d x}\left[\cos ^{-1}(\sec h x)\right]=\sec h x$
  • C
    $\frac{d}{d x}\left[\tan ^{-1}(\sin h x)\right]=\sec h x$
  • Correct answer
    D
    $\frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\sec x$

Solution

$\begin{aligned} & \text { } \frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\frac{1}{\tan ^2 \frac{x}{2}+1} \times \sec ^2 \frac{x}{2} \times \frac{1}{2} \\ & \frac{\sec ^2 \frac{x}{2}}{\sec ^2 \frac{x}{2}} \times \frac{1}{2}=\frac{1}{2}\end{aligned}$

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

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