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Question Number 65724 by bshahid010@gmail.com last updated on 02/Aug/19

Answered by mr W last updated on 03/Aug/19

$$S=\sin x+\cos x+\frac{1}{\sin x}+\frac{1}{\cos x}+\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}$$$$=\sin x+\cos x+\frac{\sin x+\cos x}{\sin x\cos x}+\frac{1}{\sin x\cos x}$$$$=\sin x+\cos x+\frac{\sin x+\cos x+1}{\sin x\cos x}$$$$let\sin x+\cos x=t$$$$\Rightarrow \sin x\cos x=\frac{t^2-1}{2}$$$$\Rightarrow S=t+\frac{2(t+1)}{t^2-1}=t+\frac{2}{t-1}=1+(t-1)+\frac{2}{t-1}$$$$since-\sqrt{2}⩽t⩽\sqrt{2}$$$$-\sqrt{2}-1⩽t-1⩽\sqrt{2}-1$$$$$$$$fort-1>0:$$$$S=1+(t-1)+\frac{2}{t-1}⩾1+2\sqrt{2}=S_{min}$$$$S_{min}att=\sqrt{2}+1>\sqrt{2}\Rightarrow impossible$$$$$$$$fort-1<0:$$$$S=1-[(1-t)+\frac{2}{1-t}]⩽1-2\sqrt{2}=S_{max}$$$$S_{max}att=1-\sqrt{2}>-\sqrt{2}\Rightarrow possible$$$$$$$$\Rightarrow mininmumof\mid S\mid is\mid 1-2\sqrt{2}\mid =2\sqrt{2}-1\approx 1.828$$$$att=1-\sqrt{2}orx=n\pi -{(-1)}^n\left({\sin }^{-1}\frac{2-\sqrt{2}}{2}\right)-\frac{\pi }{4}$$

Answered by Tanmay chaudhury last updated on 03/Aug/19

$$sinx+cosx=a$$$$\sqrt{2}(\frac{1}{\sqrt{2}}sinx+\frac{1}{\sqrt{2}}cosx)=a$$$$\sqrt{2}sin(\frac{\pi }{4}+x)=a$$$$+1⩾sin(\frac{\pi }{4}+x)⩾-1$$$$so\sqrt{2}⩾a⩾-\sqrt{2}$$$$tanx+cotx$$$$\frac{{sin}^{2}x+{cos}^{2}x}{cosxsinx}$$$$\frac{2}{2sinxcosx}$$$$\frac{2}{sin2x}$$$$\frac{2}{1+sin2x-1}$$$$\frac{2}{{(sinx+cosx)}^2-1}$$$$\frac{2}{a^2-1}$$$$secx+cosecx$$$$=\frac{sinx+cosx}{sinxcosx}$$$$=(sinx+cosx)\times \frac{1}{sinxcosx}$$$$=(sinx+cosx)\times \frac{2}{2sinxcosx}$$$$=a\times \frac{2}{a^2-1}$$$$\mathit{s}\mathit{o}\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}+\mathit{t}\mathit{a}\mathit{n}\mathit{x}+\mathit{c}\mathit{o}\mathit{t}\mathit{x}+\mathit{s}\mathit{e}\mathit{c}\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\mathit{x}$$$$=\mathit{a}+\frac{2}{{\mathit{a}}^2-1}+\frac{2\mathit{a}}{{\mathit{a}}^2-1}$$$$=\mathit{a}+\frac{2(\mathit{a}+1)}{(\mathit{a}+1)(\mathit{a}-1)}$$$$=\mathit{a}+\frac{2}{\mathit{a}-1}$$$$\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{a}=\sqrt{2}$$$$=\frac{\sqrt{2}}{1}+\frac{2}{\sqrt{2}-1}$$$$=\sqrt{2}+2(\sqrt{2}+1)$$$$=3\sqrt{2}+2$$$$=3\times 1.41+2$$$$=6.23$$$$\mathit{s}\mathit{o}\mathit{m}\mathit{o}\mathit{d}\mathit{o}\mathit{f}\mid 6.23\mid =6.23$$$$\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{a}=-\sqrt{2}$$$$=\frac{-\sqrt{2}}{1}+\frac{2}{-\sqrt{2}-1}$$$$=\frac{-\sqrt{2}}{1}-\frac{2(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}$$$$=-\sqrt{2}-2\sqrt{2}+2$$$$=2-3\sqrt{2}$$$$=-2.23$$$$\mathit{s}\mathit{o}\mathit{m}\mathit{o}\mathit{d}\mathit{o}\mathit{f}-2.23\mid -2.23\mid =2.23$$$$henceminimumvalue=2.23$$

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