in AB^Δ C prove : (( cos(A ))/a^( 3) ) +((cos(B))/b^( 3) ) +((cos(C))/c^( 3) ) ≥((81)/(16p^( 3) )) where : p= (a+b +c )/2


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Question Number 173983 by mnjuly1970 last updated on 23/Jul/22

$i n A \overset{Δ}{B C} p r o v e :$ $\frac{c o s (A)}{a^{3}} + \frac{c o s (B)}{b^{3}} + \frac{c o s (C)}{c^{3}} ⩾ \frac{81}{16 p^{3}}$ $w h e r e : p = (a + b + c) / 2$
	Answered by mnjuly1970 last updated on 23/Jul/22
	$proof : equivalently we must prove: ((cos(A))/(sin^( 3) (A))) +((cos(B))/(sin^( 3) (B))) +((cos(C))/(sin^( 3) (C))) ≥ ((81)/(2 {(sin(A)+sin(B)+sin(c))^3 ≥^? 27sin(A)sin(B)sin(C)})) also ((cos(A).sin(B).sin(C))/(sin^( 2) (A))) + ((cos(B).sin(A).sin(C))/(sin^( 2) (B))) + (( cos(C).sin(A).sin(B))/(sin^( 2) (C))) ≥(3/2) equivalently: ((cos(A).bc)/a^( 2) ) +((cos(B).ac)/b^( 2) )+((cos(C).ab)/c^( 2) )≥(3/2) also ((b^( 2) +c^( 2) −a^2 )/(2a^( 2) )) +((a^( 2) +c^( 2) −b^( 2) )/(2b^( 2) ))+((a^( 2) +b^( 2) −c^( 2) )/(2c^( 2) ))≥(3/2) equivalently: ∗{ ((b^( 2) +c^( 2) )/a^( 2) ) +((a^( 2) +c^( 2) )/b^( 2) ) +((a^( 2) +b^( 2) )/c^( 2) )} ≥^? 6 ∗ L.h.s ≥^(am−gm) (2(√((a^( 2) /b^( 2) ) .(b^( 2) /a^( 2) ))) +2(√((a^( 2) /c^( 2) ).(c^( 2) /a^( 2) ))) +2(√((b^( 2) /c^( 2) ).(c^( 2) /(b 2)) ))) L.h.s ≥ 6 ■$
	$p r o o f :$ $e q u i v a l e n t l y w e m u s t p r o v e :$ $\frac{c o s (A)}{{s i n}^{3} (A)} + \frac{c o s (B)}{{s i n}^{3} (B)} + \frac{c o s (C)}{{s i n}^{3} (C)}$ $⩾ \frac{81}{2 {{(s i n (A) + s i n (B) + s i n (c))}^{3} \overset{?}{⩾} 27 s i n (A) s i n (B) s i n (C)}}$ $a l s o$ $\frac{c o s (A) . s i n (B) . s i n (C)}{{s i n}^{2} (A)} +$ $\frac{c o s (B) . s i n (A) . s i n (C)}{{s i n}^{2} (B)} +$ $\frac{c o s (C) . s i n (A) . s i n (B)}{{s i n}^{2} (C)} ⩾ \frac{3}{2}$ $e q u i v a l e n t l y :$ $\frac{c o s (A) . b c}{a^{2}} + \frac{c o s (B) . a c}{b^{2}} + \frac{c o s (C) . a b}{c^{2}} ⩾ \frac{3}{2}$ $a l s o$ $\frac{b^{2} + c^{2} - a^{2}}{2 a^{2}} + \frac{a^{2} + c^{2} - b^{2}}{2 b^{2}} + \frac{a^{2} + b^{2} - c^{2}}{2 c^{2}} ⩾ \frac{3}{2}$ $e q u i v a l e n t l y :$ $* {\frac{b^{2} + c^{2}}{a^{2}} + \frac{a^{2} + c^{2}}{b^{2}} + \frac{a^{2} + b^{2}}{c^{2}}} \overset{?}{⩾} 6$ $* L . h . s \overset{a m - g m}{⩾} (2 \sqrt{\frac{a^{2}}{b^{2}} . \frac{b^{2}}{a^{2}}} + 2 \sqrt{\frac{a^{2}}{c^{2}} . \frac{c^{2}}{a^{2}}} + 2 \sqrt{\frac{b^{2}}{c^{2}} . \frac{c^{2}}{b 2})}$ $L . h . s ⩾ 6 ◼$
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