1.The singer waved to the people while she danced to the music.
2.The singer waved to the people while she was dancing to the music.
Are these two sentences grammatically correct?
Do they vary in meaning?
They are correct and interchangeable. However, many people would prefer the first version because "is" is a weak verb, and the second sentence is unnecessarily wordy.
"When", "while", and "as" to talk about two things that happen at the same time.
"While" is mostly used with the past continuous verb tense. It is usually used as Event A + while + Event B and to emphasis a time frame as in "during that time."
Even though both of your sentences are correct grammatically and logically, the past continuous tense is preferred to emphasis conjunction "while."
The difference is one of emphasis.
One just describes a past action: She waved to the people while she danced.
The other describes a past action where one action was ongoing then: She waved to the people while she was dancing.
During the time she was dancing, she waved to the people. In the second one, it can mean that she waved more than once.
Another example might illustrate this difference in emphasis better:
As she was walking to town, she saw the man on the bicycle.
As she walks to town, she saw the man on the bicycle.
So, do you want to just say something about a past event or do you want to express that one event was ongoing in the past?
What do you want to say?
