An object executes periodic motion which is given by sin4πt−cos  4πt.  Then we can  say that

The function sin4πt−cos  4πt will represent a periodic motion, if it is identically  repeated after a fixed interval of time. If this is also a simple harmonic motion.  Then we can uniquely write that as  Asinω t+ϕ ∴sin4πt−cas  4πt  can be reduced as follows212sin4πt−12cos  4πt (multiplying & dividing by 2 )=2cosπ4sin(4πt)−sinπ4cos  (4πt)∵sinπ4=cosπ4=12    The above relation can be written as=2sinπ  t−π4∵sinAcosB−cosAsinB=sin(A−B) The above equation can be compared with  y=Asin(ω t+ϕ) This show that the  function sin(4πt−cos  4πt) is simple harmonic with its resultant as  2sinπ  t−π4.  Its time period can be found using ω=2πT Here ω=4π,ω=4π2πT=4πT=12s