Find the values of `a` if equation `1-cosx=(sqrt(3))/2|x|x a ,x in (0,pi),` has exactly one solution.

Explanation

`1 - cos x = (sqrt(3))/(2) |x| + a` has root when `y = 1 - cos x` and `y = (sqrt(3))/(2) |x| + a` intersect.
For one real solution, consider the case when two curves touch each other.
Slope of `C_(1)` is sin x and for `x gt 0`, slope of `c_(2)` is `(sqrt(3))/(2)`. Thus, for the point of contact,
`sin x = (sqrt(3))/(2) rArr x = (pi)/(3) or (2pi)/(3)`
Hence the point of contact is `((pi)/(3), (1)/(2)) or ((2pi)/(3), (3)/(2))`
For `((pi)/(3), (1)/(2))`, we get `a = (1)/(2) - (pi)/(2sqrt(3))`
For `((2pi)/(3), (3)/(2))`, we get `a = (3)/(2) - (pi)/(sqrt(3))`