Show that (1-cot(23))(1-cot(22)) is a rational number and find its value.
Cotangent angle sum formula: cot(a+b) = [cot(a)*cot(b)-1] / [cot(a)+cot(b)]
Let b=45-a. Then cot(45) = [cot(a)*cot(45-a)-1] / [cot(a)+cot(45-a)].
cot(45)=1 so the identity can be rearranged as cot(a)*cot(45-a) -cot(a) - cot(45-a) + 1 = 2.
Factor to create the identity [cot(a)-1] * [cot(45-a)-1] = 2.
The expression given in the problem is this identity evaluated at a=23. Then (1-cot(23))(1-cot(22)) = 2.
Solution