Question

Find the value of expression ${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}$

Answer 1

Let given expression be denoted by $E$

$E={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}$

$E={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4(\pi -\frac{3\pi }{8})+{\cos }^4(\pi -\frac{\pi }{8})$

$={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{\pi }{8}$

$[\because \cos (\pi -\theta )=-\cos \theta ]$

$=2({\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8})$

$E=2({\cos }^4\frac{\pi }{8}+{\cos }^4(\frac{\pi }{2}-\frac{\pi }{8}))$

$=2({\cos }^4\frac{\pi }{8}+{\sin }^4\frac{\pi }{8})$

$[\because \cos (\frac{\pi }{2}-\theta )=\sin \theta ]$

$=2({\left({\cos }^2\frac{\pi }{8}\right)}^2+{\left({\sin }^2\frac{\pi }{8}\right)}^2)$

$=2[{({\cos }^2\frac{\pi }{8}+{\sin }^2\frac{\pi }{8})}^2-2{\cos }^2\frac{\pi }{8}{\sin }^2\frac{\pi }{8}]$

$[\because a^2+b^2=(a+b{)}^2-2ab]$

$=2-4{\cos }^2\frac{\pi }{8}{\sin }^2\frac{\pi }{8}$

$[\because {\sin }^2\theta +{\cos }^2\theta =1]$

$E=2-4{\cos }^2\frac{\pi }{8}{\sin }^2\frac{\pi }{8}$

$=2-{\sin }^2\left(\frac{\pi }{4}\right)$ $[\because \sin 2\theta =2\sin \theta \cos \theta ]$

$=2-{\left(\frac{1}{\sqrt{2}}\right)}^2$

$=2-\frac{1}{2}$

$=\frac{3}{2}$