Given `f(x) = sin^n x` , `x_n`  coordinate inflecting point `0 <x_n<pi/2` , prove `sin x_n=sqrt((n-1)/n)`  in `n>= 3` ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Since the problem provides the information that `x_n`  is the x coordinate of inflection point, hence, you need to evaluate the second order derivative such that:

`f'(x) = n*sin^(n-1) x*cos x`

You need to evaluate the second order derivative using the product rule such that:

`f''(x) = n*(n-1)*sin^(n-2) x*cos^2 x + n*sin^(n-1)x*(- sin x)`

`f''(x) = n*(n-1)*sin^(n-2) x*cos^2 x- n*sin^n x`

Factoring out `n*sin^(n-2) x`  yields:

`f''(x) = n*sin^(n-2) x((n - 1)cos^2 x - sin^2 x)`

Since the x coordinate of an inflection point is the root of the second order derivative, hence `f''(x_n) = 0`  such that:

`n*sin^(n-2) x_n((n - 1)cos^2 x_n - sin^2 x_n)`

Since the problem provides the information that `x_n in (0,pi/2), ` hence `sin x_n in (0,1), ` thus `sin^(n-2) x_n != 0`  and `(n - 1)cos^2 x_n - sin^2 x_n = 0`  such that:

`(n - 1)cos^2 x_n - sin^2 x_n = 0`

You need to use the fundamental formula of trigonometry such that:

`cos^2 x_n = 1 - sin^2 x_n`

`(n - 1)(1 - sin^2 x_n) - sin^2 x_n = 0`

You need to open the brackets such that:

`n - nsin^2 x_n - 1 + sin^2 x_n - sin^2 x_n = 0`

Reducing like terms yields:

`n - nsin^2 x_n - 1 = 0 => n - nsin^2 x_n = 1`

You need to isolate the term that contains `x_n`  to the left side such that:

`-nsin^2 x_n = 1 - n => nsin^2 x_n = n- 1`

`sin^2 x_n = (n - 1)/n => sin x_n = +-sqrt((n - 1)/n)`

You need to keep only the positive value since for `x_n in (0,pi/2), ` sin `x_n in (0,1)`  such that:

`sin x_n = +sqrt((n - 1)/n)`

Hence, evaluating `sin x_n`  for `x_n in (0,pi/2)`  and `x_n`  the `x`  coordinate of the inflection point, yields `sin x_n = +sqrt((n - 1)/n), ` hence, the given identity holds.

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