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

Minimum value of ∣ sin x + cos x + tan x + cot x + sec x + cosec x ∣ is ...

$${Minimum}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\:\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}\:+\:\mathrm{tan}\:{x}\:+\:\mathrm{cot}\:{x}\:+\:\mathrm{sec}\:{x}\:+\:\mathrm{cosec}\:{x}\:\mid\:\:\:{is}\:\:... \\ $$

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

sinx+cosx →(let sinx+cosx=a) (√2) ((1/(√2))sinx+(1/(√2))cosx) (√2) sin(x+(π/4)) (√2) ≥(√2) sin(x+(π/4))≥−(√2) (√2)≥a≥−(√2) so (√2) ≥sinx+cosx≥−(√2) tanx+cotx =(1/(sinxcosx))=(2/(sin2x))=(2/(−1+1+sin2x)) =(2/(−1+(sinx+cosx)^2 ))=(2/(a^2 −1)) secx+cosecx =(1/(cosx))+(1/(sinx)) =((2(sinx+cosx))/(sin2x)) =((2(a))/(a^2 −1)) now ∣sinx+cosx+tanx+cotx+secx+cosecx∣ =∣a+(2/(a^2 −1))+((2a)/(a^2 −1))∣ =∣a+2×((a+1)/((a+1)(a−1)))∣ =∣a+(2/(a−1))∣ put a=(√2) =∣(√2) +((2((√2) +1))/(((√2) −1)((√2) +1)))∣ =∣(√2) +2(√2) +2∣ =3(√2) +2 =6.23 put a=−(√2) ∣−(√2) +(2/(−(√2) −1))∣ ∣−(√2) −((2((√2) −1))/1)∣ ∣−(√2) −2(√2) +2∣ ∣2−3(√2) ∣ ∣−2.23∣ =2.23 put a=0 ∣a+(2/(a−1))∣ =∣0+(2/(−1))∣ =2 so minimum value is 2 pls check

$${sinx}+{cosx}\:\rightarrow\left({let}\:{sinx}+{cosx}={a}\right) \\ $$$$\sqrt{\mathrm{2}}\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}{sinx}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}{cosx}\right) \\ $$$$\sqrt{\mathrm{2}}\:{sin}\left({x}+\frac{\pi}{\mathrm{4}}\right) \\ $$$$\:\sqrt{\mathrm{2}}\:\geqslant\sqrt{\mathrm{2}}\:{sin}\left({x}+\frac{\pi}{\mathrm{4}}\right)\geqslant−\sqrt{\mathrm{2}}\: \\ $$$$\sqrt{\mathrm{2}}\geqslant{a}\geqslant−\sqrt{\mathrm{2}} \\ $$$${so}\: \\ $$$$\sqrt{\mathrm{2}}\:\geqslant{sinx}+{cosx}\geqslant−\sqrt{\mathrm{2}}\: \\ $$$${tanx}+{cotx} \\ $$$$=\frac{\mathrm{1}}{{sinxcosx}}=\frac{\mathrm{2}}{{sin}\mathrm{2}{x}}=\frac{\mathrm{2}}{−\mathrm{1}+\mathrm{1}+{sin}\mathrm{2}{x}} \\ $$$$=\frac{\mathrm{2}}{−\mathrm{1}+\left({sinx}+{cosx}\right)^{\mathrm{2}} }=\frac{\mathrm{2}}{{a}^{\mathrm{2}} −\mathrm{1}} \\ $$$${secx}+{cosecx} \\ $$$$=\frac{\mathrm{1}}{{cosx}}+\frac{\mathrm{1}}{{sinx}} \\ $$$$=\frac{\mathrm{2}\left({sinx}+{cosx}\right)}{{sin}\mathrm{2}{x}} \\ $$$$=\frac{\mathrm{2}\left({a}\right)}{{a}^{\mathrm{2}} −\mathrm{1}} \\ $$$${now}\:\mid{sinx}+{cosx}+{tanx}+{cotx}+{secx}+{cosecx}\mid \\ $$$$=\mid{a}+\frac{\mathrm{2}}{{a}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{2}{a}}{{a}^{\mathrm{2}} −\mathrm{1}}\mid \\ $$$$=\mid{a}+\mathrm{2}×\frac{{a}+\mathrm{1}}{\left({a}+\mathrm{1}\right)\left({a}−\mathrm{1}\right)}\mid \\ $$$$=\mid{a}+\frac{\mathrm{2}}{{a}−\mathrm{1}}\mid \\ $$$${put}\:{a}=\sqrt{\mathrm{2}}\: \\ $$$$=\mid\sqrt{\mathrm{2}}\:+\frac{\mathrm{2}\left(\sqrt{\mathrm{2}}\:+\mathrm{1}\right)}{\left(\sqrt{\mathrm{2}}\:−\mathrm{1}\right)\left(\sqrt{\mathrm{2}}\:+\mathrm{1}\right)}\mid \\ $$$$=\mid\sqrt{\mathrm{2}}\:+\mathrm{2}\sqrt{\mathrm{2}}\:+\mathrm{2}\mid \\ $$$$=\mathrm{3}\sqrt{\mathrm{2}}\:+\mathrm{2} \\ $$$$=\mathrm{6}.\mathrm{23} \\ $$$${put}\:{a}=−\sqrt{\mathrm{2}}\: \\ $$$$\mid−\sqrt{\mathrm{2}}\:+\frac{\mathrm{2}}{−\sqrt{\mathrm{2}}\:−\mathrm{1}}\mid \\ $$$$\mid−\sqrt{\mathrm{2}}\:−\frac{\mathrm{2}\left(\sqrt{\mathrm{2}}\:−\mathrm{1}\right)}{\mathrm{1}}\mid \\ $$$$\mid−\sqrt{\mathrm{2}}\:−\mathrm{2}\sqrt{\mathrm{2}}\:+\mathrm{2}\mid \\ $$$$\mid\mathrm{2}−\mathrm{3}\sqrt{\mathrm{2}}\:\mid \\ $$$$\mid−\mathrm{2}.\mathrm{23}\mid \\ $$$$=\mathrm{2}.\mathrm{23} \\ $$$${put}\:{a}=\mathrm{0} \\ $$$$\mid{a}+\frac{\mathrm{2}}{{a}−\mathrm{1}}\mid \\ $$$$=\mid\mathrm{0}+\frac{\mathrm{2}}{−\mathrm{1}}\mid \\ $$$$=\mathrm{2} \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{minimum}}\:\boldsymbol{{value}}\:\boldsymbol{{is}}\:\mathrm{2} \\ $$$$\boldsymbol{{pls}}\:\boldsymbol{{check}} \\ $$$$ \\ $$

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