Set up a ladder so it's safe and secure. Get a friend to hold it steady. You can easily fall off and get hurt doing this. So be careful.
Here's the later of dead leaves the palm tree sheds as it grows new leaves. This is what we will trim off.
Don't cheap out here. Get some thick leather work gloves. These are a must and will save your hands.
I use a M12 (12 volt) Milwaukee Hackzall cordless cutter. It's light, easy to manage and cuts about 30 branches per charge. You can use whatever you like. But this specific tool I feel is a must.
As you grab the leaves or branches you'll need to protect your hands. See the branches? They are as bad as they look. It's like a saw blade.
The saw blade here is key. In the M12 Milwaukee saw I use a wood/pruning blade. This one is either from Bosch or Dewalt. There are a number of brands. A fresh blade like this will but like butter.
Get secured. Working from bottom up you'll cut each branch close to the trunk like pictured. Let the branches fall to the ground. Don't try to manage it falling. You're high up and could fall.
This is what the cuts will look like. Don't worry you can go back and clean them up.
As you trim all the dead branches you might have some uneven cuts. If it bothers you then cut again to clean it up. Otherwise leaving it like pictured is just fine.