[tex]\displaystyle\\
\sin^2 135^o+\sin^2 45^o = \\\\
= \sin^2 (180^o-45^o)+\sin^2 45^o = \\\\
=\sin^2 45^o+\sin^2 45^o = \\\\
= \left(\frac{\sqrt{2} }{2}\right)^2+\left(\frac{\sqrt{2} }{2}\right)^2=\\\\\\
= \frac{2}{4} + \frac{2}{4} = \frac{2+2}{4} = \frac{4}{4} = \boxed{\bf 1}[/tex]