For n \in \mathbb{N}, let $$f_n(\theta)=\tan\left(\frac{\theta}{2}\right)\,(1+\sec\theta)(1+\sec 2\theta)(1+\sec 4\theta)\cdots(1+\sec 2^n\theta).$$ Which of the following is true?

Question

Accepted Answer

f_4(\pi/64)=1. The function f_n(θ) simplifies to 1 for specific values of θ due to the properties of the tangent and secant functions at those angles, particularly when θ is a power of 2 in radians. In this case, f_4(π/64) evaluates to 1, confirming that this option is correct.

For n \in \mathbb{N}, let \[f_n(\theta)=\tan\left(\frac{\theta}{2}\right)\,(1+\sec\theta)(1+\sec 2\theta)(1+\sec 4\theta)\cdots(1+\sec 2^n\theta).\] Which of the following is true?

Solution

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