I have adought related to this of oscullation topic how sgould we solve it

(a) sinwt - coswt .
= √2 { 1/√2 sinwt - 1/√2coswt}
= √2{cos45°.sinwt - sin45°.coswt}
= √2sin(wt - 45°)
This is in the form of y = Asin(wt ± ∅) 
So, this is the equation of SHM .
Period = 2π/w 

(b) sin³wt 
Use formula of sin3x = 3sinx - 4sin³x 
So, sin³wt = 1/4 [ 3sinwt - sin3wt ] 
Hence, you observed that this equation is combination of two SHM. Hence, this is not SHM. But periodic motion .
Period = LCM of period { sinwt , sin3wt }
Period of sinwt = 2π/w 
period of sin3wt = 2π/3w 
so, period = 2π/w