Question Number 103921 by Rio Michael last updated on 18/Jul/20
$$\mathrm{Prove}\:\mathrm{that}\:\forall\:{x}\:\in\:\bar {\mathbb{R}}\:,\:\mid\:\mathrm{cos}\:{x}\:\mid\:\leqslant\:\mathrm{1}\:−\:\mathrm{sin}^{\mathrm{2}} \:{x} \\ $$
Answered by Worm_Tail last updated on 18/Jul/20
$${cosx}=\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} {x}\:\:\:\:\:\:\:\:} \\ $$$$\:\:\:\:\mid{cosx}\mid=\:\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} {x}}\:\:\:\:\leqslant\mathrm{1}−{sin}^{\mathrm{2}} {x}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\mid{cosx}\leqslant\mathrm{1}−{sin}^{\mathrm{2}} {x}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by Rio Michael last updated on 18/Jul/20
$$\mathrm{perfect}\:\mathrm{sir}\:\mathrm{thanks} \\ $$