Can we use discrete cosine transform with imaginary part
If we have an OFDM system with $N$ subcarriers, the modulated data based on Fourier exponential function can be expressed as follows: 

$S_{ofdm} = \frac{1}{\sqrt T} e^{j2nπFt}$ where $T$ is the OFDM symbol period, $F = 1/T$ denotes the minimum spacing required to guarantee the orthogonality between subcarriers. 

Regarding the OFDM-based DCT, the modulated symbol can be done as below: 

$S_{DCT-OFDM} = \sqrt {2/T} \cos(2πnft)$  

you can compare the two equations. On one hand,  the orthogonality in DCT can be satisfied for a minimum intercarrier distance of $f = 1/(2T)$. On the second hand, the number of subcarriers resulted of DCT are double of those resulted from DFT. So, does that mean we gain or loose half spectrum efficiency?  No, that's not right, because, in DCT, you are just using the real part (or imaginary part) compared with DFT which is using complex basis function. That's all simply how we use DCT with OFDM. 

Coming back to your question, is it possible to use the real part and imaginary part?  Yes,  you can use one of them and generally, OFDM based on DCT/DFT should give (almost) similar performance.