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$\,\,\int\limits_0^{\frac{\pi }{2}} {c{\rm{o}}{{\rm{s}}^2}x.c{\rm{os}}4xdx} =
\int\limits_0^{\frac{\pi }{2}} {\frac{{1 + c{\rm{os}}2x}}{2}.c{\rm{os}}4xdx} =
\int\limits_0^{\frac{\pi }{2}} {\left( {\frac{1}{2}.c{\rm{os}}4x + \frac{1}{2}c{\rm{os}}2x\cos 4x}
\right)dx} $
$=\int\limits_{0}^{\frac{\pi}{2} } (\frac{1}{2} cos4x+\frac{1}{4} cos2x+\frac{1}{4}cos6x )dx$
$=(\frac{1}{8}sin4x+\frac{1}{4}sin2x+\frac{1}{24} sin6x)\mathop |\nolimits_0^\frac{\pi}{2}=0 $
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