I am trying to show that for every λ ∈ ( − π , π ) ∖ { 0 } and every n ∈ Z &gt; 0 1 n 1 − cos ⁡ ( n − 1 ) λ 1 − cos ⁡ λ ≤ 1

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I am trying to show that for every   λ  ∈  (  −  π  ,  π  )  ∖  {  0  } and every   n  ∈            Z              &amp;gt;      0            1    n                      1        −        cos        ⁡        (        n        −        1        )        λ                    1        −        cos        ⁡        λ              ≤  1

pressacvt · Accepted Answer

Using  cos  ⁡  x  =  1  −  2      sin    2    ⁡      x    2   for   x  ∈      R  ,      |    sin  ⁡  (  n  x  )      |    ≤  n      |    sin  ⁡  x      |   for   x  ∈      R  ,   n  ∈      N  ,it follows that      1    n                      1        −        cos        ⁡        (        n        −        1        )        λ                    1        −        cos        ⁡        λ              =      1    n        |                  sin        ⁡        (        n        −        1        )                  λ          2                            sin        ⁡                  λ          2                      |    ≤            n      −      1        n    &amp;lt;  1    .

I am trying to show that for every λ ∈ ( −<!--

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