As challenging as posting the trot may be, many riders find sitting the trot even more difficult. Here are some tips from former About Horses and Ponies forum members who will help you learn to maintain a secure seat at the trot.
Sitting the trot is a bit like trying to pat your head and rub your tummy at the same time - you have to relax your leg muscles so that your grip is not pushing your seat out of the saddle, while using your abdominal and back muscles to absorb the motion and to follow it.
You want to be sitting as deeply as possible, so do some stretching exercises before you mount - particularly a sideways lunge, where you start with feet apart and slowly transfer most of your weight to one foot, bending the knee and stretching with the other leg straight. You should feel your inner thigh muscles protest a bit. After enough repetitions, you should also feel as if your pelvis has spread a bit wider.
Once mounting, place your palms on the pommel and push yourself up out of the saddle, at the same spreading your legs in a wide "V". Lower your seat back into the saddle with your legs still stretched and feel the difference in your seat.
Next grasp the pommel (or a strap running through the "D" rings if that's more comfortable) and pull yourself even further down and forward into the saddle. Now you're ready to sit the trot, in accordance with the advice given in other posts here.
At first aim for only a few strides before repositioning yourself. Gradually add more strides, but always stop with the first bounce and regroup. This will be easier on both you and the horse.
Stop as soon as you feel yourself starting to bounce. Once you begin to bump with every stride your horse's back will stiffen in defense and the bouncing just gets worse.
Also, if you can borrow a horse that works through his back at the trot, so that the motion is more longitudinal than up and down, you'll find your "Eureka" moment will come much earlier and more easily.
At first, you just need to completely relax, like a wet noodle, and learn to follow the horse's motion. If you start to bounce, relax, breathe, let your muscles relax and follow. Resist the temptation to tighten up.
What helped a lot for me was to ride without stirrups, I found it helped to deepen my seat and improve my balance.
I usually teach the sitting trot on a lunge line. I let the rider have a strap to hold onto so they don't catch the horse in the mouth. I tell them to lean back as far as they need to and dangle their legs until they find their seat and rhythm. I sit them back up once they get it. Also, many times the sitting trot does not work because the rider is squeezing their knees and thighs; so be conscious of that the the next time you try.
Sitting trot always takes some learning to get used to. Have your trainer put you on the lunge line at first so you don't have to worry about your horse and can concentrate on your position. Start off by posting the trot and get your horse into a nice easy trot. Try lengthening your stirrups a hole or two to help relax your legs. When you start to sit, think of reverse posting. Drive your seat down into the horse on every beat instead of rising up. This will help you get the feel for the rhythm.
Make sure to remember to relax your back, seat, legs, shoulder and arms. Sit back a little farther and lean your shoulder back and open them up. This position will help you to stay balanced at the sitting trot. Try to really feel your horse's stride and push down and forward into the saddle with every one. Once you get the movement down you won't have to drive so much. Make sure you're not pinching with your knees or turned your toe out.
Bareback and no stirrup lessons will help you get the sitting trot faster.
I once had a girl having a hard time sitting the jog. I started singing a song with her latest boyfriend's name in it, she got to laughing so hard she relaxed and sat the jog. It does work, you just have to be creative.