When you accomplish something "without breaking a sweat," you do it effortlessly. You do not have to put any effort into the activity because you find it so simple. You complete the assignment swiftly or very effortlessly. You do not need to sweat.

Rahul answered all the questions without breaking a sweat.

Another expression that has the same meaning is `no sweat'.

Listen, I'll carry the two large suitcases. No sweat at all.

When you `break into a sweat', you actually start sweating. And when is it that we begin to sweat. when we strain ourselves, when we are incredibly anxious or afraid of something. The children broke into a sweat when they heard strange noises coming from the bedroom.