Question

For any natural number n , prove that inequality
|sin n x| $\ge$ n | sin x |

Answer 1

The inequality clearly holds for n = 1
We now assume
|sin mx| $\le$m | sin x | .....(1)
Now, | sin ( m + 1) x | = |sin mx cos x + cos mx sin x |
$\le$ | sin mx | | cos x | + | cos mx | | sin x |
$\le$ | sin mx | + | sin x |
[$\therefore$ | cos x | $\le$ 1 and | cos mx | $\le$ 1]
$\le$m | sin x | + | sin x | , by (1)
Thus | sin( m + 1) x | $\le$(m + 1) | | sin x |.
Hence by induction, the required inequality holds for every positive integer n