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Question Number 71984 by Henri Boucatchou last updated on 23/Oct/19

$$x,y\in I\subset \mathbb{R}provethat\mid cosx-siny\mid ⩽\mid x-y\mid$$

Answered by mind is power last updated on 23/Oct/19

$$\mathrm{x}=\mathrm{y}=0\mathrm{mistack}$$$$\Rightarrow 1⩽0$$$$\mathrm{i}\mathrm{think}\mathrm{is}\mid \sin \left(\mathrm{x}\right)-\sin \left(\mathrm{y}\right)\mid ⩽\mid \mathrm{x}-\mathrm{y}\mid$$$$\mathrm{letf}(\mathrm{x},\mathrm{y})={\int }_{\mathrm{x}}^{\mathrm{y}}\cos \left(\mathrm{t}\right)\mathrm{dt}$$$$\mathrm{f}(\mathrm{x},\mathrm{y})={\left[\sin \left(\mathrm{t}\right)\right]}_{\mathrm{x}}^{\mathrm{y}}=\sin \left(\mathrm{y}\right)-\sin \left(\mathrm{x}\right)$$$$\mid \mathrm{f}(\mathrm{x},\mathrm{y})\mid =\mid {\int }_{\mathrm{x}}^{\mathrm{y}}\cos \left(\mathrm{t}\right)\mathrm{dt}\mid ⩽\mid {\int }_{\mathrm{x}}^{\mathrm{y}}\mid \cos \left(\mathrm{t}\right)\mathrm{dt}\mid \mid ⩽\mid {\int }_{\mathrm{x}}^{\mathrm{y}}1\mathrm{dt}\mid =\mid \mathrm{y}-\mathrm{x}\mid$$$$\Rightarrow \mid \sin \left(\mathrm{y}\right)-\sin \left(\mathrm{x}\right)\mid ⩽\mid \mathrm{y}-\mathrm{x}\mid$$

Commented by Henri Boucatchou last updated on 23/Oct/19

$$U^{\prime }rerightSir,usingcosorsinonlyis$$$$available.$$

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