1. He didn't bother to let me know where she lived.
2. He didn't bother letting me know where she lived.
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Are both OK? Do we use both a to infinitive and a gerund after 'bother'?
Both might be used. I prefer #1 as better, but, in everyday speech, #2 might be used. The meaning is clear either way.
Both sentences are grammatically correct, but there is a slight difference in meaning.
In the first sentence, "He didn't bother to let me know where she lived," the phrase "to let me know" uses the infinitive form of the verb "let." This suggests that the action of not letting you know was intentional or purposeful.
In the second sentence, "He didn't bother letting me know where she lived," the phrase "letting me know" uses the gerund form of the verb "let." This suggests that the action of not letting you know was more passive or incidental.
So, to answer your question, both the infinitive form ("to let") and the gerund form ("letting") can be used after the verb "bother," but they may convey slightly different meanings. The choice between the two forms depends on the intended nuance or emphasis in the sentence.