cos-x-sin-x-2tan-x-1-cos-x-2-0-

Question Number 135679 by liberty last updated on 15/Mar/21

(\cos x - \sin x) (2 \tan x + \frac{1}{\cos x}) + 2 = 0

Answered by EDWIN88 last updated on 15/Mar/21

Let \tan \frac{x}{2} = t \to {\begin{cases} \cos x = \frac{1 - t}{1 + t^{2}} \\ \sin x = \frac{2 t}{1 + t^{2}} \\ \tan x = \frac{2 t}{1 - t^{2}} \end{cases}

\Leftrightarrow (\frac{1 - t^{2}}{1 + t^{2}} - \frac{2 t}{1 + t^{2}}) (\frac{4 t}{1 - t^{2}} + \frac{1 + t^{2}}{1 - t}) + 2 = 0

\Rightarrow (1 - t^{2} - 2 t) . (t^{2} + 4 t + 1) + 2 (1 - t^{4}) = 0

\Rightarrow {3 t}^{4} - t^{2} + {6 t}^{3} - 2 t + {9 t}^{2} - 3 = 0

\Rightarrow t^{2} ({3 t}^{2} - 1) + 2 t ({3 t}^{2} - 1) + 3 ({3 t}^{2} - 1) = 0

\Rightarrow ({3 t}^{2} - 1) (t^{2} + 2 t + 3) = 0

{\begin{cases} t^{2} + 2 t + 3 = 0 \to has complex roots \\ {3 t}^{2} - 1 = 0 \Rightarrow t = \pm \frac{1}{\sqrt{3}} \end{cases}

\Leftrightarrow \tan (\frac{x}{2}) = \pm \frac{1}{\sqrt{3}}; \frac{x}{2} = n π \pm \frac{π}{6}

\Rightarrow x = 2 n π \pm \frac{π}{3}; n \in Z

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