Find the smallest positive values of `xa n dy` satisfying `x-y=pi/4a n dcotx+coty=2`

Explanation

Given, `x-y=pi/4` (i)
`cot x+cot y =2` (ii)
From Eq. (ii), `sin (x+y)=2 sin x. sin y`
`=cos (x-y)-cos (x+y)`
`= "cos" pi/4-cos (x+y)`
or `sin (x+y)+cos(x+y)= "cos" pi/4=1/sqrt(2)`
or `1/sqrt(2) sin (x+y) +1/sqrt(2) cos (x+y)=1/2`
or `cos (x+y- pi/4)= "cos" pi/3`
`rArr x+y- pi/4 = 2n pi pm pi/3, n in Z`
or `x+y=2n pi pm pi/3+pi/4` (iii)
for `n=0, x+y=(7 pi)/12" "( :' x, y gt 0)` (iv)
From (i) and (iv), `x=(5pi)/12, y=pi/6`
Hence, the least positive values of x and y are `(5pi)/12` and `pi/6`, respectively.