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I've found the period of this trigonometric function, $$y=\sin^n(x)+\cos^n(x)$$.

when n ($n\neq2$)is odd, the period is $2\pi$,

when n is even, the period is $\frac{\pi}{2}$.

but how to proof it?

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    $\begingroup$ What about when $n=2$? $\endgroup$ – Mike Apr 24 '14 at 10:29
  • $\begingroup$ This can easily be generalized for $\sin^ax+\cos^bx$. $\endgroup$ – Lucian Apr 24 '14 at 11:20
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Consider the periods of both terms, the period of the sum is the least common multiple of both.

You could express the sine and cosine using complex exponentials, often that makes trigonometric identities trivial.

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