How do you use the sum and difference formula to simplify ##cot ((113 pi)/ 12)##?

Start by finding a coterminal radian for ##(113pi)/12## since it is a big number and we can't look it up on the unit circle. The least positive co-terminal angle in radians will be ##(13pi)/12##

Now find two radians added or subtracted will give you the above value. I have found ##(3pi)/4+(pi)/3## will give you ##(13pi)/12##

Next recall the sum identity which states ##tan(x+y)=(tanx+tany)/(1-tanxtany)##

Let's rewrite this identity in to cotangent: ##cot(x+y)=(cotx+coty)/(cotxcoty-1)##

Now substitute those two radians.

##cot((3pi)/4+pi/3)=(cot((3pi)/4)+cot((pi)/3))/(cot((3pi)/4)cot((pi)/3)-1)##

Now simplify.

##=(-1+1/sqrt(3))/((-1)(1/sqrt(3))-1)##

##=(-sqrt(3)+1)/(-1-sqrt(3))##

That's it.