Q: Is it possible to lose fat and lose muscle at the same time?

A: Yes, losing muscle and fat at the same time is completely common and normal. Whenever your body is in caloric deficit for a prolonged period of time (in other words, when you are using more calories than you are taking in), it will usually burn both muscle and fat for energy so you'd be losing both simultaneously. The best way to hang on to the muscle is to make sure you exercise it, primarily through resistance training. This basically tells your body: I need that muscle, don't use it as energy source! However if you undereat and don't weight train, then losing fat along with muscle will be the most common result. The proportion of fat and muscle lost will depend on a host of variables, including your age, health, diet, and activity level.

To understand better why your body chooses to burn both fat and muscle and not just fat for energy when undereating, consider that muscle requires energy just to maintain, while fat does not. So your body is short on calories and is basically faced with the choice: do I use the precious few calories to maintain my existing muscle mass which I do not need for survival or do I burn this muscle, get a little energy out of it and also lower my caloric demands for the future? Now it becomes clear why the body readily burns muscle when faced with a calorie deficit -- it both gives it some immediate energy (though less than burning the same amount of fat) AND it reduces the caloric requirements to maintain the existing muscle mass (since there is less muscle now). And while having a good muscle mass lets us look good and burn fat, it is certainly not a requirement for survival and most people can potentially get by on a fraction of the muscle mass they currently have.