The Telecare Medical Information System (TMIS) is a technology used in Wireless Body Area Networks (WBAN) that is used efficiently for remote healthcare services. TMIS services can be provided as cloud computing services for storage and processing purposes. TMIS uses wearable sensors to collect patient data and transmit it to the controller node over a public channel. The data is then obtained from the controller node by the medical server and stored in the database for analysis. However, an attacker can attempt to launch attacks on data transferred across an unsecured channel. Several schemes have therefore been proposed to provide mutual authentication however, there are security and performance problems. Therefore, the research aims to design two secure and efficient inter-BAN authentication protocols for WBAN: protocol-I (P-I) for emergency authentication and protocol-II (P-II) for periodic authentication. To analyze the proposed protocols, we conduct an informal security analysis, implement Burrows-Abadi-Needham (BAN) logic analysis, validate the proposed protocols using the Automated Validation of Internet Security Protocols and Applications (AVISPA) simulation tool, and conduct a performance analysis. Consequently, we show that the proposed protocols meet all the security requirements in this research, achieve mutual authentication, prevent passive and active attacks, and have suitable performance for WBAN.