Question

Prove that sin^m x + cos^m x < 1 for all natural n greater than 1.​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

The inequality clearly holds for n = 1

We now assume  

|sin mx| ≤m | sin x |            .....(1)

Now, | sin ( m + 1) x | = |sin mx cos x + cos mx sin x |

≤ | sin mx | | cos x | + | cos mx | | sin x |

≤ | sin mx | + | sin x |

             [∴ | cos x | ≤ 1 and | cos mx | ≤ 1]

≤m | sin x | + | sin x | ,              by (1)

Thus | sin( m + 1) x | ≤(m + 1) | | sin x |.

Hence by induction, the required inequality holds for every positive integer n .