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

Guest Apr 20, 2019

edited by
Guest
Apr 20, 2019

#1**+1 **

\(0 < a < \dfrac \pi 2 \Rightarrow 0 < \sin(a) < 1,~0 < \cos(a) < 1\\ 0<\sin(a)<1 \Rightarrow \sin^k(a) < \sin^j(a),~k>j\geq 1\\ 0<\cos(a)<1 \Rightarrow \cos^k(a) < \cos^j(a),~k>j\geq 1\)

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

.Rom Apr 20, 2019