Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help

cos (4pi/3 – 3pi/4)

This is not a statistics question.

You want the cosine of 240 - 135 degrees = 105 degrees. That equals -cos 75 degrees

One of the answers to your previous questions explained how to do that.
Treat 75 deg as 45 + 30 degrees, and use the cos (A + B) equation.