In this paper it is proved that any two given Legendre associated functions P n m ( μ ) P_n^m(\mu ) and P n s ( μ ) P_n^s(\mu ) , where n ⩾ 1 n \geqslant 1 is an integer and where one of the integers m or s may be 0 (and m ≠ ± s m \ne \pm s ), have either no zero in common or exactly one common zero, namely μ = 0 \mu = 0 . An auxiliary result states that the n − m n - m zeros of P n m P_n^m known to lie in the open interval ( − 1 , 1 ) ( - 1,1) lie in fact in the open interval ( − c , c ) ( - c,c) , where ± c \pm c are the two zeros of n ( n + 1 ) − m 2 / ( 1 − μ 2 ) n(n + 1) - {m^2}/(1 - {\mu ^2}) which is one of the coefficients in the Legendre associated equation satisfied by P n m P_n^m . Some monotonicity behavior of P n m P_n^m is simultaneously described. The proof of the main result is based on properties of Prüfer polar coordinates.