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![\displaystyle Se~arata~usor~ca~cos \frac{\pi}{12} =cos( \frac{\pi}{3} -\frac{\pi}{4})= \frac{\sqrt{6}+ \sqrt{2} }{4} \\ cos \frac{5\pi}{12} =cos( \frac{\pi}{2} -\frac{\pi}{12})=sin \frac{\pi}{12}= sin( \frac{\pi}{3} -\frac{\pi}{4})=\frac{\sqrt{6}- \sqrt{2} }{4} \\ A=cos^8(\frac{5\pi}{12})-cos^8(\frac{\pi}{12})=\\ =(\frac{\sqrt{6}- \sqrt{2} }{4} )^8-(\frac{\sqrt{6}+ \sqrt{2} }{4} )^8=\\ = \frac{(\sqrt{6}- \sqrt{2})^8-(\sqrt{6}+\sqrt{2})^8}{2^{16}} =\\ =\frac{4^4[(2- \sqrt{3})^4-(2+\sqrt{3})^4]}{2^{16}} ](https://tex.z-dn.net/?f=%5Cdisplaystyle+Se%7Earata%7Eusor%7Eca%7Ecos+%5Cfrac%7B%5Cpi%7D%7B12%7D+%3Dcos%28+%5Cfrac%7B%5Cpi%7D%7B3%7D+-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D+%5Cfrac%7B%5Csqrt%7B6%7D%2B+%5Csqrt%7B2%7D+%7D%7B4%7D+%5C%5C%0Acos+%5Cfrac%7B5%5Cpi%7D%7B12%7D+%3Dcos%28+%5Cfrac%7B%5Cpi%7D%7B2%7D+-%5Cfrac%7B%5Cpi%7D%7B12%7D%29%3Dsin+%5Cfrac%7B%5Cpi%7D%7B12%7D%3D+sin%28+%5Cfrac%7B%5Cpi%7D%7B3%7D+-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D%5Cfrac%7B%5Csqrt%7B6%7D-+%5Csqrt%7B2%7D+%7D%7B4%7D+%5C%5C%0AA%3Dcos%5E8%28%5Cfrac%7B5%5Cpi%7D%7B12%7D%29-cos%5E8%28%5Cfrac%7B%5Cpi%7D%7B12%7D%29%3D%5C%5C%0A%3D%28%5Cfrac%7B%5Csqrt%7B6%7D-+%5Csqrt%7B2%7D+%7D%7B4%7D+%29%5E8-%28%5Cfrac%7B%5Csqrt%7B6%7D%2B+%5Csqrt%7B2%7D+%7D%7B4%7D+%29%5E8%3D%5C%5C%0A%3D+%5Cfrac%7B%28%5Csqrt%7B6%7D-+%5Csqrt%7B2%7D%29%5E8-%28%5Csqrt%7B6%7D%2B%5Csqrt%7B2%7D%29%5E8%7D%7B2%5E%7B16%7D%7D+%3D%5C%5C%0A%3D%5Cfrac%7B4%5E4%5B%282-+%5Csqrt%7B3%7D%29%5E4-%282%2B%5Csqrt%7B3%7D%29%5E4%5D%7D%7B2%5E%7B16%7D%7D+%0A%0A "\displaystyle Se~arata~usor~ca~cos \frac{\pi}{12} =cos( \frac{\pi}{3} -\frac{\pi}{4})= \frac{\sqrt{6}+ \sqrt{2} }{4} \\ cos \frac{5\pi}{12} =cos( \frac{\pi}{2} -\frac{\pi}{12})=sin \frac{\pi}{12}= sin( \frac{\pi}{3} -\frac{\pi}{4})=\frac{\sqrt{6}- \sqrt{2} }{4} \\ A=cos^8(\frac{5\pi}{12})-cos^8(\frac{\pi}{12})=\\ =(\frac{\sqrt{6}- \sqrt{2} }{4} )^8-(\frac{\sqrt{6}+ \sqrt{2} }{4} )^8=\\ = \frac{(\sqrt{6}- \sqrt{2})^8-(\sqrt{6}+\sqrt{2})^8}{2^{16}} =\\ =\frac{4^4[(2- \sqrt{3})^4-(2+\sqrt{3})^4]}{2^{16}} ")
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