It's simple! Cognitive ethnography feeds distributed cognition theory by providing the body of observed phenomena that the theory must explain!
 

Distributed cognition provides a vastly different approach for thinking about design and how it can support HCI. Distributed Cognition is strongly positioned to better understand interactions between people and technologies than the Cognitive theories of early HCI.

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At it's very core, Distributed Cognition is used to try to understand how cognitive systems are organized. However, this theory is different from other cognitive theories in two main ways. First, Distributed Cognition looks for cognitive processes wherever they may occur within a network of elements that work together in some type of process. Second, Distributed Cognition assumes that all events don’t necessarily have to be contained within the individual. 

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Distributed cognition is also broader than many previous theories, and includes phenomena that emerge in social interactions and interactions between people and structure in their environments. Distributed Cognition also assumes that the study of cognition can't be separated from the study of culture, because agents live in complex cultural environments. Cognition is no longer isolated from culture or separate from it.

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Ethnography plays a very crucial part in distributed cognition, and this is why participant observation is a very important component of cognitive ethnography. Cognitive ethnography underlies Distributed Cognition by providing the body of observed phenomena that the theory explains. Design enters this arena in various ways. First, ethnography gives designers information about how getting things done that can enhance a design.  Trials can refine the theory of distributed cognition which in turn can be applied to improve design. Finally, since the design process creates new tools for workplaces, there are new structures and interactions to study.