replacing ceiling fan pull light switch

I have to replace the pull light switch in my ceiling fan. The ceiling fan is a Hampton Bay (the St. Claire) with light fixture. Upon disassembly I see that the pull light switch has 3 wires (blue, red and black). The blue and red wire are attached to other wires with a wire nut in the same area. The black wire runs up into the fan area. The replacement switch has blue, red and black wires that are approximately 5 in. long ... the blue and red wire are long enough but the black wire is too short to run up into the fan area. Question: can i just cut the black wire from the old switch and attach the black wire from the new switch to it with a wire nut? My concern is that I not do anything that would be a fire hazard. Thank you.

 

yes
is too short for the forum so here is some more letters.

 

thanks joed
and here are 26 more characters

 

There are an awful lot of characters in this thread.