Question

Let \[ f_n(x) = \tan\frac{x}{2}(1+\sec x)(1+\sec 2x)\cdots(1+\sec 2^{n-1}x), \] then:

• A. \( f_3\!\left(\frac{\pi}{16}\right) = 1 \)
• B. \( f_4\!\left(\frac{\pi}{16}\right) = 1 \)
• C. \( f_5\!\left(\frac{\pi}{16}\right) = 1 \)
• D. \( f_2\!\left(\frac{\pi}{16}\right) = 1 \)

Hint:

For trig telescoping products: \begin{itemize} \item Convert sec terms into cosine ratios. \item Look for cancellation patterns. \end{itemize}

Answer 1

The Correct Option is B