The period of the function f(x)=sinπ4[x]+cosπ2x+cotπ3[x]

sinπ4(8+[x])=sinπ4[x],  so its period is 8. The period of cosπ2x  is 4 and cotπ3[3+[x])=cotπ3[x]  is 3.Hence period of f(x)  is L.c.m(8,4,3)=24.