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PT
$\Leftrightarrow \cos3x +\sin7x =1-\cos(\dfrac{\pi}{2} +5x) -\left ( 1+\cos 9x\right )$
$\Leftrightarrow \cos3x +\sin7x =-\cos(\pi-(\dfrac{\pi}{2}-5x ))-\cos 9x$
$\Leftrightarrow \cos3x +\sin7x =\cos(\dfrac{\pi}{2}-5x ) -\cos 9x$
$\Leftrightarrow \cos3x +\sin7x =\sin 5x-\cos 9x$
$\Leftrightarrow \cos3x +\cos 9x=\sin 5x-\sin7x $
$\Leftrightarrow 2\cos6x \cos 3x=-2\cos6x \sin x $
$\Leftrightarrow \left[ {\begin{matrix} \cos6x =0\\ \cos3x =\sin(-x) \end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} \cos6x =0\\ \cos3x =\cos(\dfrac{\pi}{2}+x) \end{matrix}} \right. $
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