Question

Find ${\cos }^4\left(\pi /8\right)+{\cos }^4\left(3\pi /8\right)+{\cos }^4\left(5\pi /8\right)+{\cos }^4\left(7\pi /8\right)-3/2$

Answer 1

${\cos }^4\left(\pi /8\right)+$${\cos }^4\left(3\pi /8\right)+$${\cos }^4(\pi -\frac{3\pi }{8})+$${\cos }^4(\pi -\frac{\pi }{8})-3/2$

$={\cos }^4\left(\pi /8\right)+$${\cos }^4\left(3\pi /8\right)+$${\cos }^4\left(\pi /8\right)+$${\cos }^4\left(3\pi /8\right)-3/2$

$=2{\cos }^4\left(\pi /8\right)+$$2{\cos }^4\left(3\pi /8\right)-3/2$

$=2\left[{\cos }^4\right(\pi /8)+$${\cos }^4\left(3\pi /8\right)]-3/2$

$\frac{\pi }{8}=(22\frac{1}{2}{)}^{\circ }$

$\frac{3\pi }{8}=(66\frac{1}{2}{)}^{\circ }$

puting values

$=2\left[\right(0.9238{)}^4+$$\left(0.38268{)}^4\right]-3/2$

$=2[0.72855+0.02144]-1.5$

$=1.5-1.5$

$=0$