π is the period of the function

The period of |sin x| + |cos x| and sin4x + cos4x is π2. sin (sin x) + sin (cos x) has period 2π. The function 1+2cos⁡xsin⁡x(2+sec⁡x) can be written in a simplified form as cos⁡xsin⁡x=cot⁡x, so it has period π.