The inner bias tape corners on my costumes always had the highest proportion of “omg how did u do that??“s by far. Learning this black magic will make some jRPG and anime designs significantly more approachable. It’s probably best if you know how to do straight bias tape and outer corners first, since those are much easier. There are many tutorials available on the internet, including some by me, which will show you how to do these.
Similar to the process for applying bias tape to an outer corner, the first step is to line your strip of bias tape up and sew right up to the point pictured above. Fold your bias tape so that it makes a 90 degree angle, and sew up to the point again from the other side. Be precise!
At this point, the back of your fabric should look like this. CAREFULLY clip to the point at a 45 degree angle. Do NOT cut the bias tape, just the fabric you are attaching it to.
The next part is the most difficult to visualize, so I took photos from every angle imaginable. Before, your bias tape was folded and your fabric was flat. Now you want it the other way around: your fabric will be folded while your bias tape lays flat.
Bring your bias tape into alignment with itself so that it runs straight up and down, as shown in the photos. It should look basically the same on both sides. The picture taken from the bottom should help illustrate how it’s folded—the seam allowances should all be facing away from the strip of bias tape. Pin them in place.
Next, take a ruler and extend the diagonal line formed by the blue fabric on down as if it continued through your bias tape. (This is the lightly penned diagonal line.) Then, draw the 45 degree angles like you would for an outer corner, but in the \/\ formation pictured above. All these ups and downs are basically letting you sew the miter seam from the inside so it looks awesome. Stitch along the line as shown, then clip excess fabric.
Flip everything to the inside and congrats! You’re done. Finish the back with your method of choice. (Slipstitching, glue, topstitching, etc.)
Final pro-tip: these 90 degree/right angle corners are the easiest to visualize, because you get to the 90 degree angle inset by making three line segments at 45 degree angles. (This can even be reduced to two 45 degree line segments if you allow the final outer bias tape fold to stay closed, as shown in the “outer corner” tutorial.) However, obtuse and acute angles are similarly achievable with just a few alterations to the above process - namely, your seam angles will be either greater or less than 45 degrees, proportional to whatever your desired corner angle is.