Prove-that-27-sin-4-2cos-4-16-sin2-2-

Question Number 126098 by ZiYangLee last updated on 17/Dec/20

Prove that

27 (\sin^{4} α + {2 \cos}^{4} α) ⩾ 16 {(\sin 2 α)}^{2}

Answered by MJS_new last updated on 17/Dec/20

\sin 2 α = 2 \sin α \cos α

27 (s^{4} + 2 c^{4}) ⩾ {64 s}^{2} c^{2}

c = \sqrt{1 - s^{2}}

27 (3 s^{4} - 4 s^{2} + 2) ⩾ 64 s^{2} (1 - s^{2})

s^{4} - \frac{172}{145} s^{2} + \frac{54}{145} ⩾ 0

{it}^{'} s easy to show this has no real zeros

and for s = 0 {it}^{'} s true \Rightarrow always true

Commented by talminator2856791 last updated on 17/Dec/20

very nice . quadratic is perfect .