# Solve the following equation: <mtable columnalign="right left right left right left right left r

Solve the following equation: $\begin{array}{r}{\mathrm{sin}}^{14}x+{\mathrm{cos}}^{14}x=\frac{169}{64}{\mathrm{cos}}^{6}2x\end{array}$
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Paxton James
$\begin{array}{rl}2{\mathrm{sin}}^{2}x& =1-\mathrm{cos}2x\\ 2{\mathrm{cos}}^{2}x& =1+\mathrm{cos}2x\\ 128\left({\mathrm{sin}}^{14}x+{\mathrm{cos}}^{14}x\right)& =\left(1-\mathrm{cos}2x{\right)}^{7}+\left(1+\mathrm{cos}2x{\right)}^{7}\\ & =2\left(1+21{\mathrm{cos}}^{2}2x+35{\mathrm{cos}}^{4}2x+7{\mathrm{cos}}^{6}2x\right)\\ 169{\mathrm{cos}}^{6}2x& =1+21{\mathrm{cos}}^{2}2x+35{\mathrm{cos}}^{4}2x+7{\mathrm{cos}}^{6}2x\\ 0& =162{\mathrm{cos}}^{6}2x-35{\mathrm{cos}}^{4}2x-21{\mathrm{cos}}^{2}2x-1\\ 0& =\left(2{\mathrm{cos}}^{2}2x-1\right)\left(81{\mathrm{cos}}^{4}2x+23{\mathrm{cos}}^{2}2x+1\right)\\ {\mathrm{cos}}^{2}2x& =\frac{1}{2}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\frac{-23±\sqrt{{23}^{2}-4\left(81\right)}}{2\left(81\right)}<0\end{array}$