Why would you use DCT as oppose to DFT for detecting oscillation in a signal?
Just keepin' it $\Re$eal.

The DFT records the timing offset of a particular sinusoid frequency as the phase of the complex coefficient.  Since the DFT of a real signal is conjugate-symmetric, the corresponding negative frequency coefficient gets the same magnitude but negative phase.  FFT routines frequently skip calculating the negative spectrum for real signals.
This leaves NFFT/2+1 non-redundant complex coefficients, however the DC and Nyquist bins are completely real.  It would be possible to combine the DC and Nyquist bin together, making zero growth in data size (for even length signals anyway).

But this kind of flipping between Complex and Real structure may not be convenient for analysis or computation.

The DCT captures essentially the same information about the real signal, except the timing of the sinusoid is encoded as the sum of cosine waves.
The DCT has the advantage of representing a unitary transform without mixing real and complex numbers.  The DFT is also unitary, but it maps $\mathbb{R^N} \rightarrow \mathbb{C^N}$.