There are both chemical and physical methods to create the effects you are looking for.
If I understand your question, you want to achieve two effects: have the "tail" of the cape expand backward from the body of the costume, and have the bottom of the tail spread out so as to create a triangular shape.
I think both effects can be achieved by the same stiffening process.
Chemical Methods, e.g., glue or starch
The following chemical stiffening methods will require you to pre-shape the tail as you want it, then apply the glue or stiffening spray to set the shape.
Aileens.com has directions for using Aileen's Stiffen Quik.
…Stiffen-Quik Fabric Stiffening Spray comes in a convenient spray bottle for quick and easy application. The spray pump helps minimize messes and prevents overusing the adhesive so you get the most out of your bottle. Stiffen-Quik’s clear-drying formula preserves the shape and color of your project and can be layered for a heavier hold that washes out when you want to clean and reshape your fabrics. It’s perfect for seasonal fabric projects like cheesecloth ghosts and so much more!
BobVila.com has directions for making your own stiffener from glue and water.
If you need to stiffen fabric for a household project—crafting a window cornice or a new lampshade, perhaps — try using white glue and water to create an easy and inexpensive fabric stiffener. Mix equal amounts of white glue and water together in a bowl until the solution has the appearance and consistency of milk. Dip the fabric into the bowl, or paint the liquid onto the fabric. Use a form to shape the fabric as desired (or create your own form using chicken wire or heavy-duty aluminum foil). The glue will dry clear and stiff; you can apply more than one coat if the fabric needs additional stiffening.
Physical Methods, e.g., iron-on interfacing, crinoline material, tinfoil, or chicken wire
Perhaps the easiest physical technique would be to use iron-on interfacing available by the yard at any fabric store. You would iron it to the underside of the pre-shaped (laid out in the shape you want) cape tail.
This interfacting only comes in white or black, so you may need to use white and then spray paint to match the costume fabric.
You could also make a long stiff crinoline "petticoat" and attach it just under the waist section of the costume.
You can purchase crinoline material at most fabric stores. “A crinoline is a stiffened or structured petticoat designed to hold out a woman's skirt, popular at various times since the mid-19th century.” (Wikipedia)
You would use the crinoline as a long bustle, a structure worn under Victorian ladies’ dresses to create (ahem) a sort of large rump, but in your case, you would make the bustle the length of your costume tail. https://www.youtube.com/watch?v=C0FmqB62cJc
You may need to run a wire through the bottom edge of the crinoline, spread out the bottom of the crinoline along the wire, and tack it by the corners to the corners of the tail to create the triangle effect. (Similarly, if all you want is to spread out the bottom of the tail, you can run a wire through the hem of the cape.)
Use as many layers of crinoline material as you need to get the effect you want, and then loosely attach the top of the crinoline underneath the top of your cape either with a stapler or simple basting stitches. To see an example of basting stitches: http://www.the-tailoress.com/even-basting/
You may need to arrange the costume fabric to fold over the long edges of the crinoline/bustle, or you could spray paint the crinoline/bustle to match the costume fabric.
You could also create a permanent tinfoil or chicken wire underlining (form) for the tail, but these materials won't work if you plan to sit down.
Your best bet is probably to use one of the glue/spray chemical methods, or the iron-on interfacing.
You can layer the solution or spray till you get the stiffness you want, and if you sit down the worst that will happen might be a few wrinkles.
With the interfacing all you need is an iron.
Good luck, it’s a cool project.