Question Number 146868 by gsk2684 last updated on 16/Jul/21
$${if}\:\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{sec}\:{x}={a}+\mathrm{1}\:{has}\:{at}\:{least}\: \\ $$$${one}\:{solution}\:{then}\:{find}\:{the}\:{complete}\:{set} \\ $$$${of}\:{values}\:{of}\:\:'{a}'? \\ $$
Answered by Olaf_Thorendsen last updated on 16/Jul/21
$$\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{sec}{x}\:=\:{a}+\mathrm{1}\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{1}+\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{sec}{x}\:=\:{a}+\mathrm{2} \\ $$$$\mathrm{sec}^{\mathrm{2}} {x}+\mathrm{sec}{x}−\left({a}+\mathrm{2}\right)\:=\:\mathrm{0} \\ $$$$\Delta\:=\:\mathrm{1}^{\mathrm{2}} −\mathrm{4}×\mathrm{1}\left(−\left({a}+\mathrm{2}\right)\:=\:\mathrm{4}{a}+\mathrm{9}\right. \\ $$$$\left(\mathrm{1}\right)\:\mathrm{has}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{if}\:\Delta\geqslant\mathrm{0} \\ $$$$\Rightarrow\:{a}\in\left[−\frac{\mathrm{9}}{\mathrm{4}};+\infty\left[\right.\right. \\ $$
Commented by gsk2684 last updated on 16/Jul/21
$${thank}\:{you}\:{for}\:{your}\:{concern}\:{sir}. \\ $$$${please}\:{verify}\:{the}\:{bounds}\:{of}\:\mathrm{sec}\:{x} \\ $$$${i}.{e}.\:\mathrm{sec}\:{x}\:\leqslant\:−\mathrm{1}\:{or}\:\mathrm{sec}\:{x}\:\geqslant\:\mathrm{1} \\ $$