Question

sinπ/5sin2π/5sin3π/5sin4π/5=5/16 please prove stepby step​

Answer 1

Step-by-step explanation:

We have,

$\sin({ \frac{\pi}{5} } ) \sin(\frac{2\pi}{5}) \sin( \frac{3\pi}{5} ) \sin( \frac{4\pi}{5} )$

$= \sin( \frac{\pi}{5} ) \sin( \frac{2\pi}{5} ) \sin( \pi - \frac{2\pi}{5} ) \sin(\pi - \frac{\pi}{5} )$

$= \sin^{2} ( \frac{\pi}{5} ). \sin ^{2} ( \frac{2\pi}{5} )$

$= \{ \sin( \frac{\pi}{5} ) \sin( \frac{2\pi}{5} ) \}^{2}$

$= \frac{1}{4} . \{ 2\sin( \frac{\pi}{5} ) \sin( \frac{2\pi}{5} ) \} ^{2}$

$= \frac{1}{4} . \{ \cos( \frac{\pi}{5} ) - \cos( \frac{3\pi}{5} ) \} ^{2}$

$= \frac{1}{4} . \{ \cos( \frac{\pi}{5} ) - \cos( \frac{2\pi}{5} ) \} ^{2}$

$= \frac{1}{4} . \{ \frac{ \sqrt{5} + 1}{4} - \frac{ 1 - \sqrt{5} }{4} \}^{2}$

$= \frac{1}{4} . \{ \frac{ \sqrt{5} }{2} \}^{2}$

$= \frac{ 5}{16}$