Compute the Discrete Cosine Transform
Jelly, 15 bytes

LḶµ+.×þ÷L-*Ḟ³æ.


Try it online!

This uses the identity that cos(pi * x) = real(e^(i * pi * x)) = real((-1)^x)

Explanation

LḶµ+.×þ÷L-*Ḟ³æ.  Input: array A
L                Length
 Ḷ               Lowered range - [0, 1, ..., len(A)-1]
  µ              Begin a new monadic chain
   +.            Add 0.5
     ×þ          Outer product using multiplication
       ÷         Divide by
        L        Length
         -*      Compute (-1)^x
           Ḟ     Real part
             æ.  Dot product with
            ³    The first argument (A)