That's one way.
Or you can use the unit circle (if you don't know what that is search it on google).
Pressing cos(75deg) on the calculator is the easiest way.
But that's not the way you're asked to do. You are asked to do it using trigonometry.
You are on compound angles formulae, so that is the way you are supposed to do.
cos(75deg)
= cos(30deg +45deg)
= cos(30deg)cos(45deg) -sin(30deg)sin(45deg)
= [(sqrt(3) /2)*(1 / sqrt(2))] -(1/2)(1 / sqrt(2))
= [sqrt(3) / 2sqrt(2)] -[1 / 2sqrt(2)]
= [sqrt(3) -1] / [2sqrt(2)]
= [sqrt(6) -sqrt(2)] / 4
= 0.258819045 -------------------answer.