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One way it's useful is in the application of phasors in physics. Phasors require you to add up vectors of two different angles, so the cosine angle addition formula can be used if you already know the cosine and sine of the original vector angles.
You could find the exact value of cos of 15 degrees   by  using cos(45 - 30):-

cos(45 - 30) = cos 45cos30 + sin45 sin30  = 1 /sqrt2 * sqrt3/2 +  1/sqrt2* 1/2 

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