How about combining the trig terms, before solving?
restart:
eq1 := cos(lambda*ln(r1))*cos(lambda*ln(r2))
+ sin(lambda*ln(r1))*sin(lambda*ln(r2)) - 1 = 0:
combine(eq1);
cos(lambda ln(r1) - lambda ln(r2)) - 1 = 0
solve( combine(eq1), lambda, allsolutions );
2 Pi _Z1~
---------------
ln(r1) - ln(r2)
You could also utilize those assumptions. For example, using them at the combining stage, before solving.
alt := combine(eq1) assuming r1>0, r2>r1;
r1
alt := cos(lambda ln(----)) - 1 = 0
r2
solve( alt, lambda, allsolutions );
2 Pi _Z3~
---------
r1
ln(----)
r2
Or you could use them afterwards, for this particular problem.
solve( combine(eq1), lambda, allsolutions ):
combine(%) assuming r1>0, r2>r1;
2 Pi _Z5~
- ---------
r2
ln(----)
r1
And you could also do it this way,
solve( combine(eq1), lambda, allsolutions ) assuming r1>0, r2>r1;