# Complete the identity. (cos 4x-cos 8x)/(cos 4x+ cos 8x)

Complete the identity. $\frac{\mathrm{cos}4x-\mathrm{cos}8x}{\mathrm{cos}4x+\mathrm{cos}8x}$
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Brenton Gay
Complete the identity. $\frac{\mathrm{cos}4x-\mathrm{cos}8x}{\mathrm{cos}4x+\mathrm{cos}8x}$
$\mathrm{cos}p+\mathrm{cos}q=2\mathrm{cos}\frac{1}{2}\left(p+q\right)\mathrm{cos}\frac{1}{2}\left(p-q\right)$
$\mathrm{cos}p-\mathrm{cos}q=-2\mathrm{sin}\frac{1}{2}\left(p+q\right)\mathrm{sin}\frac{1}{2}\left(p-q\right)$
$\mathrm{sin}\left(-x\right)=-\mathrm{sin}x$
$\mathrm{cos}\left(-x\right)=\mathrm{cos}x$
$\frac{\mathrm{cos}4x-\mathrm{cos}8x}{\mathrm{cos}4x+\mathrm{cos}8x}=\frac{-2\mathrm{sin}\frac{1}{2}\left(4x+8x\right)\mathrm{sin}\frac{1}{2}\left(4x-8x\right)}{2\mathrm{cos}\frac{1}{2}\left(4x+8x\right)\mathrm{cos}\frac{1}{2}\left(4x-8x\right)}=\frac{\mathrm{sin}6x\mathrm{sin}2x}{\mathrm{cos}6x\mathrm{cos}2x}=\mathrm{tan}6x\ast \mathrm{tan}2x$