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I am not sure of the use of "but rather". For example, in this sentence

The trigonometric functions are important in calculus, not so much because of ..., but rather because of the properties they possess as functions.

If I am to write such a sentence, I think I may not say "rather" after "but". I would simply say "but because of the properties ....".

When to most appropriately say "but rather" is thus my question.

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The use of the adverb "rather" after "but" is also correct. It has been used emphatically in the sense of "more accurately".

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