<s>[INST] Show that $\sin^7 x + \cos^7 x < 1$ if $0 < x < \frac{\pi}{2}$. Could anybody help me solve this Problem? I don't quite get it.
This is what I got.
$\sin^7x+\cos^7x<1$
$0<x<\frac{\pi}{2}$
$0<\cos x$
$1>\sin x$
What else?
 [/INST] Using the Pythagorean theorem on the unit circle, we can easily figure out the identity of $\sin^2 x + \cos^2 x = 1$. We also know that if $0<x<\frac{\pi}{2}$, then $0 < \sin x < 1$ and $0 < \cos x < 1$. We know that if $0<\alpha<1$ and $n>1$ are true, then $\alpha ^n < 1$. If $\sin^2 x + \cos^2 x = 1$, then $\boxed{\sin^7 x + \cos^7 x < 1}$
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