?$(1)$?證明：∵?$AC=CD$?
∴?$\widehat{AC}=\widehat{CD}$?
∴?$∠CAD=∠ABC$?
∵?$AB$?是直徑
∴?$∠ACB=90°$?
∴?$∠ABC+∠BAC=90°$?
∵?$AE=AF，$??$∠ACB=90°$?
∴?$∠FAC=∠CAD，$??$CE=FC$?
∴?$∠FAC=∠CAD=∠ABC$?
∴?$∠FAC+∠CAB=90°$?
∴?$AF⊥AB$?
又∵?$AB$?是直徑
∴?$AF $?是?$⊙O$?的切線
?$ (2) $?連接?$BD$?
∵?$cos ∠ABC=cos∠CAD=\frac {AC}{AE}=\frac {4}{5}，$??$AC=4$?
∴?$AE=5$?
∴?$AF=AE=5，$??$CE=\sqrt{AE^2-AC^2}=\sqrt{5^2-4^2}=3$?
∴?$CE=CF=3$?
∵?$tan F = \frac {AC}{CF}= \frac {AB}{AF}$?
∴?$ \frac {4}{3} =\frac {AB}{5}$?
∴?$AB=\frac {20}{3}$?
∵?$cos∠ABC=\frac {4}{5}=\frac {AB}{BF}$?
∴?$BF=\frac {25}{3}$?
∴?$ BE=BF-EF=\frac {25}{3} -(3+3)=\frac {7}{3}$?
∵?$∠CAD=∠CBD，$??$∠AEC=∠BED$?
∴?$△ACE∽△BDE$?
∴?$\frac {AE}{BE}=\frac {EC}{ED} $?
∴?$\frac {5}{\frac {7}{3}}= \frac {3}{ED} $?
∴?$ED=\frac {7}{5}$?
∴?$AD=ED+AE=\frac {7}{5} +5=\frac {32}{5}$?
∴?$cos ∠BAD= \frac {AD}{AB} =\frac {\frac {32}{5}}{\frac {20}{3}} =\frac {24}{25}$?
