anprime =Cos[t] an-Sin[t] cn;
bnprime=Cos[t] bn-Sin[t] dn;
cnprime=Sin[t] an+Cos[t] cn;
dnprime=Sin[t]bn+Cos[t] dn;

InExpression=(anprime^2)+(bnprime^2)+(cnprime^2)+(dnprime^2);
Simplify[InExpression]

JnExpression=anprime dnprime - bnprime cnprime;
Simplify[JnExpression]

a1prime = Cos[t] a1 - Sin[t] c1;
b1prime = Cos[t] b1 - Sin[t] d1;
c1prime = Sin[t] a1 + Cos[t] c1;
d1prime = Sin[t] b1 + Cos[t] d1;
KnExpressionPrime=((a1prime^2)+(b1prime^2))((anprime^2)+(bnprime^2))+((c1prime^2)+(d1prime^2))((cnprime^2)+(dnprime^2))+2(a1prime c1prime+b1prime d1prime)( anprime cnprime+bnprime dnprime);
KnExpression=((a1^2)+(b1^2))((an^2)+(bn^2))+((c1^2)+(d1^2))((cn^2)+(dn^2))+2(a1 c1+b1 d1)( an cn+bn dn);
Simplify[KnExpressionPrime - KnExpression]