I understand that if a compound has a plane of symmetry it loses its optical activity, it is very easy to understand this when we have four different substituents on a single carbon atom but when we have larger more complicated structures for example rings with many substituents which are themselves stereogenic how can we be so sure it wont have any enantiomers if we only find one fortunate plane which produces a perfect mirror image?
That's really simple. Imagine two different vessels which are precise mirror images of each other; so are all the molecules inside. Imagine that you shine polarized light on them. Being the mirror images of each other, they have to rotate light in opposite directions. But as the compound itself is symmetric, they are in fact identical, so they have to rotate light in the same direction. How can these conditions be reconciled? Just like that: don't rotate light at all.
So it goes.