Question

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

• A. $\frac{2}{3}$
• B. $-\frac{3}{2}$
• C. $\frac{3}{2}$
• D. $-\frac{2}{3}$

Answer 1

The correct option is C $\frac{3}{2}$

Given ${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^{4}7\frac{\pi }{8}={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4(\frac{\pi }{2}+\frac{\pi }{8})+{\cos }^4(\frac{\pi }{2}+\frac{3\pi }{8})$

$={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\sin }^4\frac{\pi }{8}+{\sin }^4\frac{3\pi }{8}[\because \cos (\frac{\pi }{2}+\theta )=-\sin \theta ]$

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

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

$=1-\frac{1}{2}{\left(2\sin \frac{\pi }{8}\cos \frac{\pi }{8}\right)}^2+1-\frac{1}{2}{\left(2\sin \frac{3\pi }{8}\cos \frac{3\pi }{8}\right)}^2$

$=2-\frac{1}{2}{\sin }^2\frac{\pi }{4}-\frac{1}{2}{\sin }^2\frac{3\pi }{4}=2-\frac{1}{2}\times {\left(\frac{1}{\sqrt{2}}\right)}^2-\frac{1}{2}\times {\left(\frac{1}{\sqrt{2}}\right)}^2[\because \sin \frac{3\pi }{4}=\sin (\pi -\frac{\pi }{4})=\sin \frac{\pi }{4}]$

$=\frac{3}{2}$