There are a couple of ways to enter Ganon's Castle without requiring the rainbow bridge. This can be useful if you only want to get the Golden Gauntlets, for example, but without the Light Arrows you will not be able to kill Ganondorf.
This method uses Contortion Hovering to cross the gap, starting at the top of the broken arch. As you enter the area, there is a path leading up the mountain on the right which enables you to reach the starting position. Eight bombchus are required to make the distance.
Usually, once all the silver rupees in the Fire Trial have been collected, one has to Longshot to the platform in order to proceed. However, if you do a hoverslide from the platform with the Fire Slug on and perform either a jumpslash or a backflip (jumpslash is easier to time) then it is possible to make the distance without using the Longshot. Once you jumpslash, hold backwards to ensure you run safely onto the platform.
Align to the red ice, before moving to the approximate position in the video. As soon as the freezzards appear on the screen, sidehop to the left, then take out your Hookshot and move to the right. When you reach the freezzards, use the hookshot and if the positioning is correct, you will clip through. From there, quickspin them to kill them, which unlocks the door.
To begin with, the most efficient way to get up to the switch covered by red ice is to do a precise jump from the block. For the set up and angle, follow what is done in the video.
To hit the switch through the red ice, get right into the corner and do a jump attack with the Megaton Hammer.