No, in Taekwondo, the red belt is not the same rank as a black belt. Depending on your school, it is either one or two steps below the black belt. Some schools use a belt that is half red and half black to designate a black belt who is under age. This is called a poom.
In taekwondo, the red belt is commonly associated with the 2nd grade, that is, the level just before black belt. All taekwondo schools have their own colored belt systems, so some schools might use a brown belt instead of a red belt. Some schools might use both a brown belt and a red belt.
This differs from the use of the red belt in karate. In karate, the red belt represents the tenth degree black belt.
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In the traditional Okinawa styles of karate, a red and white belt represents 7th and 8th Dan degrees. The 9th and 10th are a solid red belt.