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In this paper, we investigate the capacity gains offered by cognitive radio in a spectrum-sharing system where the transmit power and rate of the secondary user are adjusted based on the channel variations of the secondary link and spectrum-sensing information pertaining to the activity of the licensed user. We assume a primary/secondary spectrum-sharing system where the secondary users may have access to the spectrum band originally assigned to the primary (licensed) user, as long as the interference power inflicted on the primary receiver is considered unharmful. In this context, considering joint average interference-power and peak transmit-power constraints, we first obtain the optimal power allocation scheme, namely variable power, for maximizing the achievable capacity of the secondary user over fading channels. Thereafter, we look into the variable rate and power adaptation policy by maximizing the achievable capacity under said power constraints and bit error rate requirements in multilevel quadrature amplitude modulation (M-QAM). Finally, the benefits of using soft-sensing information about the primary user's activity on the power and rate adaptation strategies are assessed, and numerical results and comparisons illustrating the performance of our spectrum-sharing system in different operating scenarios are provided.