No.
You actually have both cores on...or to better phrase it...both cores can be turned on rather quickly. At any given moment, your second core is turned on and goes to like 900MHz then drops back to off. Then it may come on and jump up to 1500 MHz and then drop to 900 MHz, and then turn off again.
It's more efficient to run one core all the time and the second core only as needed than it is to run both cores on at the same time all the time. If you ever watch your CPU frequencies in real time, you'll see this in action. It's cool to watch. If you've never seen this in action, then you probably assume that a 1.5GHz core means it runs at 1.5GHz. That's not true. It can run as fast as 1.5GHz, but it doesn't run that fast all the time. It only uses what it needs. It can go as low as 384MHz and as high as 1512 MHz and many spots in between.
So if your screen was on, and you were doing nothing except look at your wallpaper, your core should be somewhere between 384 and 1000 MHz...bouncing back and forth between those numbers. You'd even see an occassional jump to 1500. If you had both cores on, they would both stay between 384 and 1000 most of the time, but you'd probably still see one of them bounce up to 1500 at some point before coming back down.
In the end, there's no real reason to have a core on if there aren't enough computations taking place to need that core to be on. Keeping both cores on full time is probably more likely to contribute to extra wear and tear...at least for the second core and by creating more heat overall.