Periodicity of trigonometric identities

[tough pivot]

how to find the periodicity of |sin x| + |cos x|
both sin x and cos x has an individual periodicity of pi
I know that we are supposed to take LCM of both periodicity
isn't the LCM of pi and pi = pi ?

[surreal bough]

try visualising the individual functions and what you would get when you add them together

because I can tell you the periodicity is not pi

my thinking geometrically is that sin and cos on their own are 2π periodic, and sin(π/2-x)=cosx, so you know that cosx is essentially a π/2 shift of sinx

from here you know that the modulus of cos and sin will flip up everything that is below y=0, so the peaks of cosx will be where the troughs of sinx will be, when added together you will get a period of π/2

[tough pivot]

thanks @surreal bough