Verified answer

Let S = sin and C = cos.

tan + cot = S/C +C/S = (S^2+C^2)/(SC) = 1/(SC)

tan/csc + sin/tan = (S/C)S + S(C/S) = S^2/C+C = (S^2+C^2)/C = 1/C

Generally you can try to expand everything in terms of C and S, then simplify. You can stop when you can't simplify anymore.