why does the distance between a crest and a trough is half the wavelength??? please tell why it is like that and give proof....

Wavelength is ditance between two points at same positions. So basically distance between two crest is wavelength. And one crest and trough is formed alternatively. So thats why the distance is half.

Wavelength is the distance between any two consecutive similar points on a wave, by similar as in it’s position wrt propogation axes, it’s velocity and its acceleration and other quantities.
let

be the equation of the snapshot of the wave.
Now a consecutive crest and trought will have opposite displacements
let distance between them be d
so we have

WLOG the initial distance be x=0 so we get

which implies the minimum kd is = \pi
hence

Hence proved