How are you using the inhalers? Do you have a spacer device or do you use them directly into the mouth? Have you been shown the right way to use them ?
The blue inhaler is mainly to help you if you have problems with the breathing. It is termed a 'reliever' and should improve your breathing after a few minutes.
A brown inhaler is a steroid, mainly given to reduce inflammation.
You are missing the 'preventer', often a green inhaler but could be something else. Sometimes a purple inhaler is given, which is a 'combi' - combining the green and brown. Speak to your GP about this, I'm not saying you must have the preventer, they may have reasons for not using one in your case - 'no two people are the same' of course! - but the info I have is that they should not be giving the brown one without. "Steroid inhalers are only used in combination with a long-acting beta-agonist inhaler (this can be with two separate inhalers or with a single inhaler containing two medicines)" from patient.co.uk/health/inhale... (perhaps worth a read)
Using a spacer device is supposed to help the medication get into the lungs better by dispersing the dose into a 'cloud' that you inhale via the spacer, where as a direct blast into the throat can leave half the stuff on your tongue or on the back of your throat, where it gets swallowed, rather than breathing it in. There should be advice given to rinse your mouth after using the inhalers too.
I'm taking it you've been for regular check-ups and they've ruled out anything else that could be causing the coughing? The campaign from earlier this year about '...if you have had a cough more than 3 weeks...' springs to mind.
You could do to be speaking with a COPD nurse and check that you are using the inhalers correctly. Call the BLF helpline, 10am to 6pm weekdays, on 03000 030 555. Have a chat with the nurse and see what they can suggest.
Medication is supposed to make you feel better, not worse. It could be that you don't get on with the brown inhaler, your GP can perhaps change to a different one to see if that cures it. I went through that phase as I was getting a dry, sore throat with mine and I was working on the radio at the time, so couldn't do with having a bad throat like that. We changed the inhaler type a few times before settling on the one I have now.
Hope that helps. If it were me I'd experiment a bit, and see what happens for a day or two of not using the brown one.