Let f(x)=12(|sinx|cosx+sinx|cosx|) .The period of f(x)  is 2π

On the interval [0,2π],  the function can be represented, as follows f(x)={tanx       x∈[0,π2)0               x∈(π/2,π]−tanx    x∈[π,3π/2)0                x∈[3π/2,2π] Since the period of tanx  is π  so f  is periodic with period 2π.