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This paper studies wireless spectrum sharing between a primary radio (PR) link and a secondary radio or so-called cognitive radio (CR) link. Assuming that the PR adapts its transmit power and/or rate upon receiving an interference signal from the CR, and such transmit adaptations are observable by the CR, this phenomenon leads to a new form of feedback from PR to CR, termed as hidden PR feedback, whereby the CR obtains the information of PR transmit adaptations without the need of a dedicated feedback channel from the PR. This interesting interaction between PR and CR is investigated in this paper to design new learning and transmission strategies for the CR, named as active learning and supervised transmission, respectively. For the active learning, this paper shows that via exploiting the hidden PR feedback, the CR is able to estimate the channel gain from the CR transmitter to the PR receiver, which is essential for the CR to predict the resulting PR performance degradation as a function of the CR transmit power level. Moreover, this paper shows that the hidden PR feedback enables the CR to design a supervised transmission to effectively control the "feedback" interference from the PR by properly setting the CR transmit power level. This paper analyzes the CR achievable rate with/without the operation of canceling the PR feedback interference at the CR receiver.