Periodomania.

ohh my god .... I don't believe I got 225 Point's for this atmost Level-2 , Problem.......

Method 1: If you draw both |sin(x)| and |cosx| separately you will see that the modulus function flips the negative part of the curve, such that each curve has a period of $\pi$ . Now both curves are symmetrical and periodic, but they are in antiphase so the period of |sinx| + |cosx| is $\frac{\pi}{2}$

Method 2 (probably the better one): Draw a unit circles and cut off the bottom part. If you pick a point of the semicircle at angle x and drop a perpendicular line to form a right-angled triangle, then the adjacent side is equal to cos(x) and the opposite will be equal to sin(x). Now the only quadrant that satisfies both sin(x) > 0 and cos(x) > 0 is the top-right quadrant. Therefore the period is $\frac{\pi}{2}$ .

Feel free to ask for clarification :)