Question

cos(pi/8)*cos(7pi/8)*cos(3pi/8)*cos(5pi/8)

Answer 1

$$\begin{array}{l} \cos \frac {\pi}{8} * \cos \frac {7 \pi}{8} * \cos \frac {3 \pi}{8} * \cos \frac {5 \pi}{8} = \cos \frac {\pi}{8} * \cos \left(\pi - \frac {\pi}{8}\right) * \cos \frac {3 \pi}{8} * \cos \left(\pi - \frac {3 \pi}{8}\right) \\ = \cos \frac {\pi}{8} * (- \cos \frac {\pi}{8}) * \cos \frac {3 \pi}{8} * (- \cos \frac {3 \pi}{8}) = \cos^ {2} \frac {\pi}{8} * \cos^ {2} \frac {3 \pi}{8} \\ = \left(\cos \frac {\pi}{8} * \cos \frac {3 \pi}{8}\right) ^ {2} = \left(\frac {1}{2} \left[ \cos \left(\frac {\pi}{8} + \frac {3 \pi}{8}\right) + \cos \left(\frac {3 \pi}{8} - \frac {\pi}{8}\right) \right]\right) ^ {2} \\ = \frac {1}{4} \left(\cos \left(\frac {\pi}{2}\right) + \cos \left(\frac {\pi}{4}\right)\right) ^ {2} = \frac {1}{4} * \frac {1}{2} = \frac {1}{8} \\ \end{array}$$