Cosmos is a distributed database system used for any purpose that a DB can be used for. Data Lake is a "Big data" solution that is designed for storing and processing analytical data. It isn't usable as a traditional DB.
U-SQL is a database system that can be used for any purpose, but the purpose of the system that uses the SQL database is what determines overall purpose. Data lake is designed to collect and store data for analysis at a later date, which allows for business intelligence apps to operate.
Cosmos DB is a database infrastructure system. It can store data of any type and allows creation of apps for any purpose. Those apps store data necessary to function in the DB. cosmos makes that data available in a highly redundant, distributed manner.
Cosmos db can be used to store analytical data just like data lake. The main difference is that cosmos can store other type of data for other purposes.
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