redundancy of sin and cos waves with real data
Both sin and cos waves are needed. Sure, you can find a signal (somewhere between a sine and a cosine, which seems as if it can be evaluated using either a sine or a cosine but in the general case you need both waves. And FT needs both waves anyway because internally it works with complex numbers regardless of the input  you pass to it (real input is usually interpreted as real0, img0, real1, img1...)

Suppose you have a pure sine wave and want to estimate its spectral content. Which wave would you use, a sine or a cosine?

It should be evident that only sine waves can accurately estimate the frequency and amplitude of your tested sine signal (provided that they're in phase and the frequency resolution is narrow enough to reduce spectral leakage effetcs). If you now inspect the DFT/FFT results, you'll see that only a single, imaginary coefficient (corresponding to the sine wave being tested) contributes to the overall amplitude/magnitude.  Likewise, cosine waves are necessary to accurately estimate the cosine components of your signal. Since most real signals are neither sin or cos waves, you need to use both to estimate them properly.

In addition to that, the DC component can only be represented by using a cosine wave (since sin(DC) = 0)

I personally don't think that you need to fully understand the math behind FT to realize that both waves are necessary.