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I think it’s more about how this doesn’t follow the usual rule for adjective agreement. By that rule it would be le manteau blanc / la veste blance, but that sounds weird and un-French I suppose so “blanche” it is. (I don’t know French very well so take this all with a grain of salt)
That’s essentially it. It’s a general rule that adjectives ending in -c turn into either -che or -cque.