${\cos }^{2}5°+{\cos }^{2}10°+{\cos }^{2}15°+\cdots +{\cos }^{2}360°=$

${\cos }^{2}5°+{\cos }^{2}10°+{\cos }^{2}15°+\cdots +{\cos }^{2}360°$

$=1-{\sin }^{2}5+1-{\sin }^{2}10+\cdots +1-{\sin }^{2}360$

$=(1+1+\cdots 72\text{\hspace{0.17em}times})-({\sin }^{2}5+{\sin }^{2}10+{\sin }^{2}15+\cdots +{\sin }^{2}360)$

$=72-(18+18)$

$=72-36$

$=36$