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Question Number 65192 by naka3546 last updated on 26/Jul/19

$$Minimumvalueof$$$$\mid \sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec}x\mid is...$$

Answered by Tanmay chaudhury last updated on 26/Jul/19

$$sinx+cosx\to (letsinx+cosx=a)$$$$\sqrt{2}(\frac{1}{\sqrt{2}}sinx+\frac{1}{\sqrt{2}}cosx)$$$$\sqrt{2}sin(x+\frac{\pi }{4})$$$$\sqrt{2}⩾\sqrt{2}sin(x+\frac{\pi }{4})⩾-\sqrt{2}$$$$\sqrt{2}⩾a⩾-\sqrt{2}$$$$so$$$$\sqrt{2}⩾sinx+cosx⩾-\sqrt{2}$$$$tanx+cotx$$$$=\frac{1}{sinxcosx}=\frac{2}{sin2x}=\frac{2}{-1+1+sin2x}$$$$=\frac{2}{-1+{(sinx+cosx)}^2}=\frac{2}{a^2-1}$$$$secx+cosecx$$$$=\frac{1}{cosx}+\frac{1}{sinx}$$$$=\frac{2(sinx+cosx)}{sin2x}$$$$=\frac{2\left(a\right)}{a^2-1}$$$$now\mid sinx+cosx+tanx+cotx+secx+cosecx\mid$$$$=\mid a+\frac{2}{a^2-1}+\frac{2a}{a^2-1}\mid$$$$=\mid a+2\times \frac{a+1}{(a+1)(a-1)}\mid$$$$=\mid a+\frac{2}{a-1}\mid$$$$puta=\sqrt{2}$$$$=\mid \sqrt{2}+\frac{2(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)}\mid$$$$=\mid \sqrt{2}+2\sqrt{2}+2\mid$$$$=3\sqrt{2}+2$$$$=6.23$$$$puta=-\sqrt{2}$$$$\mid -\sqrt{2}+\frac{2}{-\sqrt{2}-1}\mid$$$$\mid -\sqrt{2}-\frac{2(\sqrt{2}-1)}{1}\mid$$$$\mid -\sqrt{2}-2\sqrt{2}+2\mid$$$$\mid 2-3\sqrt{2}\mid$$$$\mid -2.23\mid$$$$=2.23$$$$puta=0$$$$\mid a+\frac{2}{a-1}\mid$$$$=\mid 0+\frac{2}{-1}\mid$$$$=2$$$$\mathit{s}\mathit{o}\mathit{m}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{u}\mathit{m}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathit{i}\mathit{s}2$$$$\mathit{p}\mathit{l}\mathit{s}\mathit{c}\mathit{h}\mathit{e}\mathit{c}\mathit{k}$$$$$$

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