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Intuitively, this problem makes sense, but I am having a hard time showing it. can someone give me a hint

 

Show that \(\left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 1\) if \(0 < a < \pi/2\)

 May 1, 2024
 #1
avatar+9664 
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Note that \((\sin (a))^5 < 1\) and \((\cos(a))^5 < 1\).

Then, we have

\(\begin{array}{rcl} (\sin(a))^7 + (\cos(a))^7 &=& (\sin(a))^2(\sin(a))^5 + (\cos(a))^2 (\cos(a))^5\\ &<& (\sin(a))^2 + (\cos(a))^2\\ &=& 1 \end{array}\)

 

The inequality is shown.

 May 1, 2024

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