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Any hints/solutions are appreciated

 

QUESTION

Show that

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

 

Thanks so much <3

 Apr 20, 2019
edited by Guest  Apr 20, 2019
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\(0 < a < \dfrac \pi 2 \Rightarrow 0 < \sin(a) < 1,~0 < \cos(a) < 1\\ 0<\sin(a)<1 \Rightarrow \sin^k(a) < \sin^j(a),~k>j\geq 1\\ 0<\cos(a)<1 \Rightarrow \cos^k(a) < \cos^j(a),~k>j\geq 1\)

 

\(\sin^7(a) < \sin^2(a)\\ \cos^7(a) < \cos^2(a)\\ \sin^7(a) + \cos^7(a) < \sin^2(a) + \cos^2(a) = 1\)

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 Apr 20, 2019

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