Question

Cos^π/8 +cos^3π/8+ cos^5π/8+cos^7π/8=2

Answer 1

Answer on Question #81078 – Math – Trigonometry

Question

$$\cos^ {\wedge} \pi / 8 + \cos^ {\wedge} 3 \pi / 8 + \cos^ {\wedge} 5 \pi / 8 + \cos^ {\wedge} 7 \pi / 8 = 2$$

Solution

Since $\cos \alpha + \cos b = 2 \cdot \cos \frac{\alpha + b}{2} \cdot \cos \frac{\alpha - b}{2}$

$$\begin{array}{l} \cos \left(\frac {\pi}{8}\right) + \cos \left(\frac {3 \pi}{8}\right) + \cos \left(\frac {5 \pi}{8}\right) + \cos \left(\frac {7 \pi}{8}\right) \\ = 2 \cdot \cos \left(\frac {\frac {7 \pi}{8} + \frac {\pi}{8}}{2}\right) \cos \left(\frac {\frac {7 \pi}{8} - \frac {\pi}{8}}{2}\right) + 2 \cdot \cos \left(\frac {\frac {5 \pi}{8} + \frac {3 \pi}{8}}{2}\right) \cos \left(\frac {\frac {5 \pi}{8} - \frac {3 \pi}{8}}{2}\right) \\ = 2 \cdot \cos \left(\frac {4 \pi}{8}\right) \cos \left(\frac {3 \pi}{8}\right) + 2 \cdot \cos \left(\frac {4 \pi}{8}\right) \cos \left(\frac {\pi}{8}\right) \\ = 2 \cdot \cos \left(\frac {\pi}{2}\right) \cdot \left(\cos \left(\frac {3 \pi}{8}\right) + \cos \left(\frac {\pi}{8}\right)\right) = 0 \\ \end{array}$$

because $\cos \left(\frac{\pi}{2}\right) = 0$

Answer: $\cos (\pi /8) + \cos (3\pi /8) + \cos (5\pi /8) + \cos (7\pi /8) = 0$

Answer provided by https://www.AssignmentExpert.com