解：$(1)\frac {\mathrm {sin}30°}{\mathrm {cos}30°}=\frac 12÷\frac {\sqrt 3}2=\frac {\sqrt 3}3=\mathrm {tan}30°$
对于任意锐角α，都有$\frac {\mathrm {sin}α}{\mathrm {cos}α}=\mathrm {tan}α$
理由：如图，$\mathrm {sin}α=\frac ac，$$\mathrm {cos}α=\frac bc，$$\mathrm {tan}α=\frac ab$
∴$\frac {\mathrm {sin}α}{\mathrm {cos}α}=\frac ac÷\frac bc=\frac ab=\mathrm {tan}α$
$(2)①cos^2 45°+sin^2 45°=(\frac {\sqrt 2}2)^2+(\frac {\sqrt 2}2)^2=\frac 12+\frac 12=1$
$②cos^2 60°+sin^2 60°=(\frac 12)^2+(\frac {\sqrt 3}2)^2=\frac 14+\frac 34=1$
发现：对于任意锐角α，都有$cos^2α+sin^2α=1$
理由：如图，$\mathrm {sin}α=\frac ac，$$\mathrm {cos}α=\frac bc$
$cos^2 α+sin^2 α=(\frac bc)^2+(\frac ac)^2=\frac {a^2+b^2}{c^2}=\frac {c^2}{c^2}=1$
