Question

16cos2π/15*cos4π/15*cos8π/15*cos16π/15=1

Answer 1

Answer:

$= 16 \cos( \frac{2\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \ \cos( \frac{16\pi}{15} ) \\ \\ = 16 \cos( \frac{2\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \cos(\pi + \frac{\pi}{15} ) \\ \\ = 16 \cos( \frac{4\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \cos( \frac{\pi}{15} ) \\ \\ = \frac{ - 8}{ \sin( \frac{ x}{15} ) } \times 2\sin( \frac{\pi}{15} ) \cos( \frac{2\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \cos( \frac{\pi}{15} ) \\ \\ = \frac{ - 4}{ \sin( \frac{x}{15} ) } \times 2 \sin( \frac{2\pi}{15} ) \cos( \frac{2\pi}{15} ) \cos( \frac{2\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \\ \\ = \frac{ - 2}{ \sin( \frac{x}{15} ) } \times 2 \sin( \frac{4\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \\ \\ = \frac{ - 2}{ \sin( \frac{x}{15} ) } 2 \sin( \frac{4\pi}{15} ) \cos( \frac{4\pi}{15} ) \cos( \frac{8\pi}{15} ) \\ \\ = \frac{ - 1}{ \sin( \frac{x}{15} ) } \times 2 \sin( \frac{8\pi}{15} ) \cos( \frac{8\pi}{15} ) \\ \\ = \frac{ - 1}{ \sin( \frac{x}{15)} } \times \sin( \frac{16\pi}{15} ) \\ \\ = \frac{ - \sin(\pi + \frac{\pi}{15} ) }{ \sin( \frac{\pi}{15} ) } \\ \\ = \sin \frac{ \frac{\pi}{15} }{ \frac{\pi}{15} } = 1$