[tex]\displaystyle\\
30)~~\sin^2 80^o+\sin^2 10^o=\sin^2 80^o+\cos^2 80^o=\boxed{1}\\\\\\
31)~~\text{tg}^2\,30^o+\text{ctg}^2\,45^o=\left(\frac{\sqrt{3} }{3} \right)^2+1^2 = \frac{3}{9}+1=\frac{1}{3}+1=\boxed{\frac{4}{3}}\\\\\\
32)~~\cos 10^o +\cos 20^o +\cos 160^o +\cos 170^o = \\ \\
= \cos 10^o +\cos 20^o +\cos (180 -20) +\cos (180-10) = \\ \\
= \underline{\cos 10^o} +\underline{\underline{\cos 20^o}} -\underline{\underline{\cos (20)}} -\underline{\cos (10)} = \boxed{0}[/tex]
