Recall the following 2 triangles from Values of Trigonometric Functions:

30-60 degree triangles 45-45 degree triangles

We use the exact sine and cosine ratios from the triangles to answer the question as follows:

`{:(cos 75^"o",=cos(30^("o")+45^("o"))),(,=cos\ 30^("o")\ cos\ 45^("o")-sin\ 30^("o")\ sin\ 45^("o")),(,=sqrt3/2(1)/sqrt2-1/2(1)/sqrt2),(,=(sqrt3-1)/(2sqrt2)):}`

This is the exact value for cos 75°.