“Cost basis” in its simplest sense is the amount you paid for an asset whose value changes over time. A dollar is a dollar, but a share of stock might be purchased for $19 and later be worth $5 or may be worth $100. When assets are bought and sold, there may be capital gains tax on the difference (gain) between the cost basis and the sale price, subject to various exclusions that are written into the Internal Revenue Code. The burden of proof is always on the taxpayer, though.
At time of death, assets in which the deceased retained an interest are included in the gross value of the estate, even if they are passing to someone “outside the Will” because they are structured to have a joint owner, or are payable to a designated beneficiary (such as a T.O.D. transfer at death account at a brokerage), or it’s real estate that two people jointly owned and now the first owner has died. The basis of these assets “steps up” to date-of-death value. If a married person, for instance, acquires such assets from her spouse, one half of the basis will step up at the first death. If she later goes and sells the asset, she has to calculate the capital gains tax on the differential. if the tax return is audited, she’ll want to have back-up documentation.
So it is a good idea to appraise assets and to look up and print out the values of them at the time the first spouse or joint owner dies, so that you can save this documentation for later use. It can be very difficult to calculate the basis for such assets long after the fact. Even where the estate has everything jointly owned with the spouse and there are no estate or inheritance tax returns to file, taking the time to obtain and preserve this documentation of value can provide tax savings and time savings later on.