Can you use a focal reducer with a teleconverter? I don't believe it would be possible to do it but, is it?
If I were to use a full-frame 50mm lens on a 2x teleconverter with a 0.71x focal reducer on APS-C, would it work? Would I also get the benefit of an extra stop of light from the focal reducer?
Theoretically, optically, it would work (for low-quality values of "work").
Yes, the focal reducer would give an extra stop of light. But because the 2x teleconverter is responsible for 2 stops of light loss, you'd still net 1 stop of light loss.
Aside from the optics math, there are some real practical problems with this approach. Depending on exactly which 50mm lens(es) you're talking about, some of them have rear elements that sit quite far back. And depending on which teleconverter(s) you're talking about (especially 2x teles), the teleconverter might intrude into the lens's space. So it is very possible that a 2x tele will not mate a certain 50mm lens.
Generally, wide-angle and normal focal length lenses are not meant to be mated with teleconverters. It's tempting to think of lenses, filters, and teleconverters as sort of snap-together, Lego-like modular components that can be stacked together and added onto. Physical mating-wise, that is true. But everything you add in the optical path has drawbacks. And items that weren't specifically designed to work together will have worse combined drawbacks than items that were meant to work together.
Teleconverters have real, visible downsides (loss of light, some degree of loss of quality or sharpness, etc.). Those downsides are acceptable to turn a 200mm lens into a 280mm or 400mm lens, without having to carry such a monstrous beast (with the corresponding monstrous price). But those downsides will quickly turn a nifty 50 into a crappy 70 or 100, when those are already readily available without compromise.
Real-world teleconverter compatibility
Nikon's AF-S Teleconverter Compatibility page shows that the widest lens any of their teleconverters support is the AF-S VR Micro 105mm ƒ/2.8G IF-ED, and even then, autofocus is not possible with that lens combined with any of their telecons. The rest of the primes that are supported are all 200mm or longer. The widest supported zoom lens is the 70-200mm ƒ/2.8.
The widest lens on Wikipedia's List of compatible EF lenses for Canon's teleconverters is the 135mm ƒ/2L USM; after the 180mm ƒ/3.5L Macro USM, the rest are 200mm and longer. (Except for the tilt-shift lenses, but they are special cases).
Sigma's Teleconverter Compatibility page is slightly more confusing (because of the new teleconverters to support the newer Sports and Contemporary lenses, vs. the older telecons for their older lenses, and because of the different mounts they support). However, none of their teleconverters support Sigma prime lenses wider than 105mm.
Current Nikon and Sigma teleconverters are more-or-less brand specific — don't bother trying to mix a Nikon teleconverter with a Sigma lens, or vice-versa. See this Sigma 2.0x Teleconverter Review at photographylife.com:
One of my biggest frustrations with the Sigma 120-300mm f/2.8 Sport lens, was the fact that it will not take any of the Nikon teleconverters – they simply will not physically mount. I am not sure if there is a technical reason for not allowing that, but given that teleconverters are mostly “pass-through” lenses for magnification purposes, I initially did not think that I would be faced with any cross-brand compatibility issues. It turns out that teleconverters are very brand-specific. Nikon teleconverters are designed to only work well with Nikkor lenses, while Sigma teleconverters are designed to only work well with Sigma lenses. So if you were thinking of buying a Sigma 1.4x teleconverter to be used for a Nikon telephoto or super telephoto lens – forget about it. Even though the Sigma 1.4x will physically mount on any Nikon telephoto lens that can take teleconverters and it will perform well optically, it will cause all kinds of lens communication and autofocus issues.
Canon teleconverters are not so limited in their compatibility with third party EF lenses. See this answer to Compatibility of Canon extenders with non-Canon lenses. Also related, Can the Sigma TC-1401 1.4x Teleconverter be used with Canon lenses?
In theory, it should "work".
But the light you "gain" by using the 0.71X focal reducer will be more than offset by the light you "lose" from using the 2X teleconverter. Likewise, the focal length you "gain" by using the 2X TC will be partially offset by the focal length you "lose" by using the 0.71X focal reducer.
If your 50mm lens has a maximum aperture of f/2, then the 2X TC makes it a 100mm f/4 lens. Using the focal reducer alone makes it a 35mm f/1.4 combination. Adding both the 0.71X focal reducer and the 2X TC to the optical path changes the entire combination to 71mm f/2.8.
That's before you include the light loss due to the additional number of optical elements in the path. No lens element passes 100% of the light that falls on it through to the other side. It's also before you add in the increase in aberrations due to the use of the TC and Focal reducer. TCs in particular tend to add CA and pincushion distortion. Then throw in the fact that most TCs are optimized to be used with lenses in the moderate to long telephoto range (e.g. 135-400mm).
For pretty much all popular ILC systems it is much more practical to use a different 70mm or 100mm lens than to use a 2X TC + 0.71X reducer on a 50mm lens. You'll spend less and get better optical image quality.