Q43 / 155 · WBJEE 2025 · Mathematics

Let $f_n(x)=\tan \frac{x}{2}(1+\sec x)(1+\sec 2 x) \ldots\left(1+\sec 2^n x\right)$, then

  • A
    $f_5\left(\frac{\pi}{16}\right)=1$
  • B
    $f_4\left(\frac{\pi}{16}\right)=1$
  • C
    $f_3\left(\frac{\pi}{16}\right)=1$
  • Correct answer
    D
    $\mathrm{f}_2\left(\frac{\pi}{16}\right)=1$

Solution

$f_n(x)=\tan 2^n x, \quad f_5\left(\frac{\pi}{16}\right)=\tan \left(32 \cdot \frac{\pi}{16}\right)=0$ $\begin{aligned} & f_4\left(\frac{\pi}{16}\right)=\tan \left(16 \cdot \frac{\pi}{16}\right)=0 \\ & f_3\left(\frac{\pi}{16}\right)=\tan \left(8 \cdot \frac{\pi}{16}\right)=\text { undefined } \\ & f_2\left(\frac{\pi}{16}\right)=\tan \left(4 \cdot \frac{\pi}{16}\right)=1\end{aligned}$

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

Relevant Questions