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I`m currently creating some shape key and trying to apply them with a driver.

I want to find out $tan()$ function is available. How can I know what function that available in driver or anywhere?

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This is a very general question. The expressions section of the manual can give you a starting point for what functions can be used in drivers, especially the list at the bottom.

But you have access to any Python libraries that are installed with your version of Blender, so first you need to find out what libraries you have, and then you need to search the internet for documentation of what functions are available in those libraries.

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