So we use the toothpaste while we are fasting? The answer is yes we can be we will have to avoid the swallowing of toothpaste. Lets see what great scholars says about this?
Shaykh Ibn Baaz said:
there is nothing wrong with that so long as one is careful to avoid swallowing any of it, just as it is prescribed in Islam for the fasting person to use the siwaak (tooth-stick).
Shaykh Muhammad al-Saalih said:
it is permitted, but it is preferable not to use them because the toothpaste is too strong and may go down into the throat. Instead of doing that during the day, one should do it at night.
Here Is a Video of IRF President Dr Zakir Naik about the same question which you can watch: