Question

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

Answer 1

$co{s}^4\frac{\pi }{8}+co{s}^4\frac{3\pi }{8}+co{s}^4\frac{5\pi }{8}+co{s}^4\frac{7\pi }{8}$

$=co{s}^4\frac{\pi }{8}+co{s}^4(\frac{\pi }{2}-\frac{\pi }{8})+co{s}^4(\frac{\pi }{2}+\frac{\pi }{8})+co{s}^4(\pi -\frac{\pi }{8})$

$=co{s}^4\frac{\pi }{8}+{\left\{cos\right(\frac{\pi }{2}-\frac{\pi }{8}\left)\right\}}^4+{\left\{cos\right(\frac{\pi }{2}+\frac{\pi }{8}\left)\right\}}^4+{\left\{cos\right(\pi -\frac{\pi }{8}\left)\right\}}^4$

$=co{s}^4\frac{\pi }{8}+si{n}^4\frac{\pi }{8}+(-sin\frac{\pi }{8}{)}^4+(-cos\frac{\pi }{8}{)}^4$

$=co{s}^4\frac{\pi }{8}+si{n}^4\frac{\pi }{8}+si{n}^4\frac{\pi }{8}+co{s}^4\frac{\pi }{8}$

$=2co{s}^4\frac{\pi }{8}+2si{n}^4\frac{\pi }{8}$

$=2(co{s}^2\frac{\pi }{8}{)}^2+2(si{n}^2\frac{\pi }{8}{)}^2$

$=2(\frac{1+cos\frac{\pi }{8}}{2}{)}^2+2(\frac{1-cos2.\frac{\pi }{8}}{2}{)}^2$ $[\because cos2x=1-2si{n}^{2}x,cos2x=2co{s}^{2}x-1]$

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

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

$=\frac{2}{4}\left\{\right(1+\frac{1}{\sqrt{2}}{)}^2+(1-\frac{1}{\sqrt{2}}{)}^2\}$

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

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

$=\frac{1}{2}\times 3=\frac{3}{2}$