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Question Number 164703 by amin96 last updated on 20/Jan/22

Answered by mr W last updated on 20/Jan/22

Σ_(n=1) ^(997) (tan^2 x_n +(1/(tan^2 x_n ))) =Σ_(n=1) ^(997) (tan^2 x_n +(1/(tan^2 x_n ))−2+2) =Σ_(n=1) ^(997) {(tan x_n −(1/(tan x_n )))^2 +2} ≥Σ_(n=1) ^(997) {2}=997×2=1994 = is valid only if tan x_n −(1/(tan x_n ))=0, i.e. tan x_n =(1/(tan x_n )) ⇒tan x_n =±1 ⇒x_n =k_n π±(π/4)

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{997}} {\sum}}\left(\mathrm{tan}^{\mathrm{2}} \:{x}_{{n}} +\frac{\mathrm{1}}{\mathrm{tan}^{\mathrm{2}} \:{x}_{{n}} }\right) \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{997}} {\sum}}\left(\mathrm{tan}^{\mathrm{2}} \:{x}_{{n}} +\frac{\mathrm{1}}{\mathrm{tan}^{\mathrm{2}} \:{x}_{{n}} }−\mathrm{2}+\mathrm{2}\right) \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{997}} {\sum}}\left\{\left(\mathrm{tan}\:{x}_{{n}} −\frac{\mathrm{1}}{\mathrm{tan}\:{x}_{{n}} }\right)^{\mathrm{2}} +\mathrm{2}\right\} \\ $$$$\geqslant\underset{{n}=\mathrm{1}} {\overset{\mathrm{997}} {\sum}}\left\{\mathrm{2}\right\}=\mathrm{997}×\mathrm{2}=\mathrm{1994} \\ $$$$=\:{is}\:{valid}\:{only}\:{if}\:\mathrm{tan}\:{x}_{{n}} −\frac{\mathrm{1}}{\mathrm{tan}\:{x}_{{n}} }=\mathrm{0},\:{i}.{e}. \\ $$$$\mathrm{tan}\:{x}_{{n}} =\frac{\mathrm{1}}{\mathrm{tan}\:{x}_{{n}} }\:\Rightarrow\mathrm{tan}\:{x}_{{n}} =\pm\mathrm{1}\:\Rightarrow{x}_{{n}} ={k}_{{n}} \pi\pm\frac{\pi}{\mathrm{4}} \\ $$

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