It depends on what the other company's "trade name" registration is. If it is an assumed name (DBA), then you do not need to change the name of your business. If the other business has a federal or state trademark for the name, then you may have to change the name of your business.
Often, the registration of the trade name does create enforcable rights by the owner of the trade name, and in fact he may need to take steps to protect the trade name from improper use by another to continue to maintain the rights. . One alternative you might have is to obtain a license from the owner of the trade name to use that same name. If the markets in which you operate do not compete, that may be an acceptable solution for you both.
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Also take a look at your logos - are they similar? Would someone mistake your business for the other because of the logo or just because of the name? You may have an issue if confusion can occurr on the part of the consumer.