If n is an odd positive interger, then
`((cosA+cosB)/(sinA-sinB))^(n)+((sinA+sinB)/(cosA-cosB))^(n)=`

Explanation

LHS`=((2cos(A+B)/(2)cos(A-B)/(2))/(2cos(A+B)/(2)sin(A-B)/(2)))^(n)`
`+((2sin(A+B)/(2)cos(A-B)/(2))/(2sin(A+B)/(2)sin(B-A)/(2)))^(n)`
`=(cot(A-B)/(2))^(n)+(-cot(A-B)/(2))^(n)`
`[because sin(-theta)=-sintheta]`
`=cot^(n)(A-B)/(2)+(-1)^(n)cot(n)(A-B)/(2)`
`=cot^(n)(A-B)/(2)[1+(-1)^(n)]`
`={0, "if n is odd" 2 cot^(n)(A-B)/(2)`, if n is even.