What if it was "something you did" that triggered your child's sadness? Is that a little harder?
For instance, your child may be sad (and may express more than a little anger) about a new household rule based on your family values like, "No electronics for the summer". What then? How do you hold the "tension of opposites" as Jung would put it?
Well, consider what you might do if something you did to set a boundary or state your own need resulted in an adult - your spouse or a friend - feeling sad or disappointed. What would you do? Isn't it another one of those "Yes...And" kind of moments?
This becomes a key skill in a relationship dance - embracing both empathy (an understanding of the needs of the "other") and boundaries (clear understanding of YOUR needs). Remember, a secure relationship is a balance between each party having autonomy and connection, with the clear understanding that the parent-child relationship is NOT one of equals because you have more power by dint of your role...and with that, more responsibility.
In this case, you step in and acknowledge those feelings of sadness, agree with them (because they ARE sad), and then
offer comfort (to meet the need of sadness). This does NOT mean that you eliminate the source of their sadness by giving them their electronics! Instead, you offer an explanation for why you have made the choice you have - because you're wanting them to have an active, play filled summer break - and HOLD both truths. YES, they are sad AND this is the new rule. Offer comfort AND clarity. Always with gentle, warm presence.
Learning to hold this balance is essential in life. They will be sad and need to cope (find ways to meet the needs that sadness brings) without any resolution to the cause of sadness. Their dog will not return from "the farm"; their grade will not be changed; they will not get an invitation to the birthday party; they will not get put onto the sports team; and they will not have access to electronics for the summer.
Life can be hard, but they can learn that it doesn't also have to be lonely if you can meet their need for comfort when they are sad.