$\cos ec48°+\cos ec96°+\cos ec192°+\cos ec384°=$

$\cos ec48°+\cos ec96°+\cos ec192°+\cos ec384°$

$=\cos ec192°+\cos ec48°+\cos ec96°+\cos ec384°$

$=\cos ec\left(270°-78°\right)+\cos ec\left(90°-42°\right)+\cos ec\left(180°-84°\right)+\cos ec\left(360°+24°\right)$

$=-sec78°+sec42°+\cos ec84°+\cos ec24°$

$=\frac{1}{\cos 42°}-\frac{1}{c\mathrm{os}78°}+\frac{1}{\sin 84°}+\frac{1}{\sin 24°}$

$=\frac{2\left(\cos 78°-\cos 42°\right)}{2c\mathrm{os}78°\cos 42°}+\frac{2\left(\sin 84°+\sin 24°\right)}{2\sin 84°\sin 24°}$

$=\frac{2\left(-2\sin 60°\sin 18°\right)}{\cos 120°+\cos 36°}+\frac{2\left(2\sin 54°\cos 30°\right)}{\cos 60°-\cos 108°}$

$=\frac{-2\sqrt{3}\left(\frac{\sqrt{5}-1}{4}\right)}{{\displaystyle \frac{-1}{2}+\frac{\sqrt{5}+1}{4}}}+\frac{2\sqrt{3}\cos 36°}{{\displaystyle \frac{1}{2}+\sin 18°}}$

$=\frac{-2\sqrt{3}\left(\sqrt{5}-1\right)}{\sqrt{5}-1}+\frac{2\sqrt{3}\left(\frac{\sqrt{5}+1}{4}\right)}{{\displaystyle \frac{1}{2}+\frac{\sqrt{5}-1}{4}}}$

$=-2\sqrt{3}+\frac{2\sqrt{3}\left(\sqrt{5}+1\right)}{\sqrt{5}+1}$

$=-2\sqrt{3}+2\sqrt{3}=0$