Question

Prove that (1+ cos pieby10) (1+cos3pieby10) (1+cos7pieby10) (1+cos9pieby10)=1 by16

​

Answer 1

Step-by-step explanation:

We have,

$(1 + \cos( \frac{\pi}{10} ) )(1 + \cos( \frac{3\pi}{10} ) )(1 + \cos( \frac{7\pi}{10} ) )(1 + \cos( \frac{9\pi}{10} ) )$

$= 2 \cos^{2} ( \frac{\pi}{20} ). 2 \cos^{2} ( \frac{3\pi}{20} ).2 \cos^{2} ( \frac{7\pi}{20} ).2 \cos^{2} ( \frac{9\pi}{20} )$

$= 16.\cos^{2} ( \frac{\pi}{20}) \cos^{2} ( \frac{3\pi}{20} )\cos^{2} ( \frac{\pi}{2} - \frac{3\pi}{20} ) \cos^{2} ( \frac{\pi}{2} - \frac{\pi}{20} )$

$= 16.\cos^{2} ( \frac{\pi}{20} )\cos^{2} ( \frac{3\pi}{20} )\sin^{2} ( \frac{\pi}{20} )\sin^{2} ( \frac{3\pi}{20} )$

$= 4\cos^{2} ( \frac{\pi}{20} )\sin^{2} ( \frac{\pi}{20} ).4\cos^{2} ( \frac{3\pi}{20} )\sin^{2} ( \frac{3\pi}{20} )$

$= \sin^{2} ( \frac{\pi}{10} ) \sin^{2} ( \frac{3\pi}{10} )$

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

$= ( \frac{5 - 1}{16}) ^{2}$

$=( \frac{1}{4} )^{2}$

$= \frac{1}{16}$