How could a spouse prove adultery?
Adultery is hard to prove, and it always had been. Carried out in secret, its discovery by a wounded spouse makes for anger and rage. Proving adultery in a fault divorce creates injuries and wounds that never heal. Before the divorce is over, both the plaintiff and defendant will be bloodied and naked.
Aside from the emotional damage, proving adultery usually relies on circumstantial evidence that the party (usually the husband, in the old days) had the intent and opportunity, an allegation usually substantiated by such evidence as hotel recipients, photographic evidence, love letters, diaries, and witnesses who, if they exist, often did not want to be involved in the case for reasons that are easily understandable.
In the old days and now, a spouse accused of adultery can admit it, deny it, or do neither, and since it is a criminal offense in jurisdictions like New York, the accused did not have to respond at all because such an admission would be self-incrimination.
For sure, however, the parties to adultery tried to keep it secret, but it often became apparent to the aggrieved spouse.
And in the old days, sometimes the "other woman" would be named as co-respondent in the divorce action by which an aggrieved wife held her husband accountable for his infidelity.