The value of  ${\sin }^2\frac{2\pi }{3}+{\cos }^2\frac{5\pi }{6}-{\tan }^2\frac{3\pi }{4}=$

${\sin }^2\frac{2\pi }{3}+{\cos }^2\frac{5\pi }{6}-{\tan }^2\frac{3\pi }{4}$

$={\sin }^{2}120°+{\cos }^{2}150°-{\tan }^{2}135°$

$={\sin }^2(90°+30°)+{\cos }^2(90°+60°)-{\tan }^2(90°+45°)$

$={\cos }^{2}30°+{\sin }^{2}60°-{\cot }^{2}45$

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

$=\frac{3}{4}+\frac{3}{4}-1$

$=\frac{6}{4}-1$

$=\frac{2}{4}=\frac{1}{2}$