Prove that (n+1)cos(pi)...

[eternal]

Prove that
[tex](n+1)cos{\frac{\pi }{n+1}}[/tex][tex]-ncos\frac{\pi }{n}\ge 1[/tex] ,[tex]n\in N[/tex]

[martin123456]

Prove that
[tex](n+1)cos{\frac{\pi }{n+1}}[/tex][tex]-ncos\frac{\pi }{n}\ge 1[/tex] ,[tex]n\in N[/tex]

we have to prove that [tex]f(n)-1 \ge 0[/tex]. differentiating LHS gives [tex]\sin\frac{\pi}{n}-\sin\frac{\pi}{n+1}[/tex] that is [tex]2\sin\frac{\pi}{2n(n+1)}\cos\frac{\pi(2n+1)}{2n(n+1)}[/tex]. [tex]2n(n+1) \geq 4[/tex] => first multiple is positive. [tex]\frac{2n+1}{2n(n+1)} \leq \frac{1}{2}[/tex] for n > 1. n=1 is trivia, not considering it now. so the derivative is [tex]\geq 0[/tex] so f(n)-1 is non decreasing...easy from here