The one child that was on the deed was on the deed from the beginning when the co-op was purchased. Reason being mother needed help obtaining a mortgage which is now paid off. From my understanding he would then be 100% owner after her death. Could he then sell the co-op and split the proceeds with the other three children of the deceased without having to pay capital gains or gift taxes.?
The deed is not like some club membership. Each person has some ownership either an undivided interest (with joint tenancy) or a divided one (tenants in common).
Being on the deed doesn't tell us what we need to know. Unless the words "Joint Tenancy with Rights of Survivorship" or similar wording appear in the grantee of the deed, the ownership was a most likely a 50/50 divisible share.
It does NOT automatically transfer to the surviving tenant. The half interest by the deceased goes to the estate.
If you sell the property for more than the basis there is capital gains to be paid no matter what the deed says. The only thing that remains to be figured is what the basis is. The person on the deed retains the basis paid when the property was purchased (as adjusted for capital improvements and sales costs). There's an exclusion of up to $250,000 if the person lived there as the principal residence for 24 out of the past 60 months, but otherwise the tax is DUE.
The basis of the share owned by the deceased steps up to the value at the time of the death, but even that may be below the sales price in an appreciating market.
So, we still have not sufficient information to answer the question.