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The relationship:
(a + j b)(cos + j sin)
is known as the FFT butterfly. In its normal expansion it looks like the following:(a + j b)(cos + j sin) = (a cos - b sin) + j (a sin + b cos)
which requires a total of 4 multiplies and 2 adds.
In hardware configurations in which the multiply operation is significantly more time-consuming than the add operation, an alternate expansion looks like the following:
(a + j b)(cos + j sin) = x + j y where x = (a + b)(cos - sin) + a sin - b cos and y = a sin + b cos
which requires a total of 3 multiplies and 5 adds.