$\mathrm{The} \mathrm{value} \mathrm{of} \cos {20}^{\circ }+2{\sin }^2{55}^{\circ }-1-\sqrt{2}\sin {65}^{\circ }=$

$\cos {20}^{\circ }+2{\sin }^2{55}^{\circ }-1-\sqrt{2}\sin {65}^{\circ }=$

$=\cos 20°+1-\cos 110°-1-2\sin {65}^{\circ }\cos 45°$

$=\cos 20°+\cos (90°+20°)-\left(\sin \right({65}^{\circ }+45°)+\sin ({65}^{\circ }-45°\left)\right)$

$=\cos 20°+\sin 20°-(\sin 110°+\sin 20°)$

$=\cos 20°+\sin 20°-(\cos 20°+\sin 20°)$

$=0$