I watched the youtube video, and I don't really agree with using home cleaning products on animals (or people). It just doesn't seem safe. I certainly wouldn't use any product that I felt the need to wear gloves with. However, I have never tried it, so that is all my personal preference.
Sorry if that photo is huge. (Also not me showing here, I allowed my younger sister to ride him at this show.) I put it up to show that I do have a fair amount of experience with whitening. There are a few things you should know:
1. Horses, like humans, have varying type of hair.
2. Health affects not just their coat, but mane & tail as well.
3. Environmental factors such as climate, minerals in the feed/water and dirt type, UV exposure, living situation, etc. all affect the hair.
Horses will need different products based on the above factors. The shampoo/conditioner that work great for one horse might not work so well for another. I have an entire collection of shampoos, conditioners, sprays detanglers, etc. I will change the "mix" based on what works best for each of my horses and what seems to work best with the type of water (when using a traveling wash-rack) Straight well water is usually the hardest to work with.
If you want to get your horse from complete "yuck" to whiter than white, you will absolutely need access to hot water. If you just need to touch up a well kept horse, tepid water will do. I avoid cold-cold water at all costs because its too uncomfortable. You will also need *time* When I'm washing, I plan 60-90 minutes for ^ Snoopy. For less white horses 45-60 minutes. For a quick wash on a dark horse about 30 minutes. I always start with the tail. I also do the *whole* tail, not just the ends.
1. You need to wet it, add some shampoo and work that in, then add some more water so that the entire tail is wet and taking in the suds. If you can separate portions of the tail and it is dry anywhere in the inner layers, add more shampoo and more water. Keep agitating in sections. I then rinse. (I can easily use 1/3 to 1/2 a 32oz bottle of Shampoo for an entire body/hair deep clean on Snoopy)
2. Repeat step 1 until your shampoo suds come out completely clear of dirt. Anywhere from 1-4 times depending on the thickness, length and soil of the tail.
3. Wring out the excess water and allow to dry for a few minutes (this is when I usually wet and do the 1st shampoo of the mane)
4. Once the tail has stopped dripping, but is still damp throughout, I will put my "purple" shampoo on FULL strength. Very lightly work it though the hairs so that it spreads out and coats the tail, but do NOT make it sudsy. I then tie the tail in a knot and let it sit from 10-25 minutes. If it dries in, that's even better.
5. When the tail has mostly dried, I sprinkle some water on and then suds up the purple shampoo really good. Depending on my time, I may let the tail sit with purple suds for another 5-10 minutes.
6. Squeeze down the suds and give a very light test rinse to see if it has gotten as white as you like. If not repeat from step 3.
7. Don't worry if you can't get all of the purple out. You need purple hues to counteract any gold-tones (urine / dirt) Once its dry, you won't really see the purple. Plus, the purple tint will wear off in 2-3 days if it really bothers you. Honestly I have never been able to over-purple a tail.
8. While the tail is still damp, apply a really good conditioner, preferably one that is not full of waxy substances (mane & tail is the absolute worst product imo) Work the conditioner in completely, using sections if necessary. I then tie the tail in a knot (or 3 if its thick) and leave it for 10-15 minutes. (Bonus if you can heat it a bit. I use the blower portion of my grooming vacuum) Some tails I will brush the conditioner through. Just depends on the horse and my time level.
9. Rinse the conditioner out. Depending on the product I use, I will leave about 10% in the tail, just enough so it still feels a little slippery, but not so much that it "clogs" up or creates a gunky tail.
10. Spray the tail with a good detangler / leave in conditioner. I avoid silicone based, drying products (like showsheen) I also work in some gel detangler. (I used to exclusively use cowboy magic detangler & shine, but it started making my hands itch like mad, so I have switched to eZall detangler and shine.)
11. Comb the tail out, braid (if you aren't worried about having wavy hair) sock/bag if you want & you're done.
I will very often dry my horses using my electro-groom because it makes their hair amazingly baby soft. I will even do a "blow-out" of the mane & tail using a flat brush and the electro-groom. Tails grow best when they are kept clean and well conditioned because this prevents breakage. Dry hair also absorbs the stains. I like to spray mine pretty often with a bottle of detangler that has a blob of conditioner mixed into it to keep them soft.
Sorry for the book. I hope some of that information was helpful :)