On the simplification using trigonometric functions
If you use the Euler's formula, you can simplify like this:
$$
[d]_{k,n} = \frac{\sqrt{2}}{N}\left( \cos{\left[ \frac{(k-1)(2n-1)\pi}{2N} \right]} e^{j\frac{2 \pi nk}{N}} \right)
$$

I think we can't simplify more.  

PS: If you use your expressions of $A$ and $B$ and again the Euler's formula, you will get the same result.