Retrieve DFT coefficients from DCT coefficients A vector a is transformed via a discrete cosine transform (DCT) to give vector b. For example in Matlab
b=dct(a);.
Vector a is also transformed via a discrete Fourier transform (DFT) to give vector c. E.g.
c=fft(a);
What is the fastest way to determine c from b without using a or retrieving it? For example,
c=fft(idft(b));
is not allowed.
Behind an FFT or a DCT operator, which can be implemented as a matrix product. Depending on the shape of your discrete vectors, you may have  $b=D\times a$ and  $c=F\times a$. So you can find your answer with a suitable matrix product. You can also find fast algorithms by decomposing each of the above matrices in simpler matrix product, or using lifting-like implementation. The very fastest can be more difficult to find.