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Question Number 112145 by weltr last updated on 06/Sep/20

$$4^{\tan {}^{2}x}+2^{\frac{1}{\cos {}^{2}x}}-80=0$$

Answered by bemath last updated on 06/Sep/20

$$4^{\tan {}^{2}x}+2^{\sec {}^{2}x}-80=0$$$$4^{\tan {}^{2}x}+2^{\tan {}^{2}x+1}-80=0$$$$set{2}^{\tan {}^{2}x}=q$$$$\Rightarrow q^2+2q-80=0$$$$\Rightarrow (q+10)(q-8)=0$$$$\Rightarrow q=8\Rightarrow 2^{\tan {}^{2}x}=2^3\Rightarrow \tan x=\pm \sqrt{3}$$

Commented by weltr last updated on 06/Sep/20

$$isthisthefinalanswer?$$

Commented by bobhans last updated on 06/Sep/20

$$\tan \mathrm{x}=\sqrt{3}\to \mathrm{x}=60°+\mathrm{k}.180°$$$$\tan \mathrm{x}=-\sqrt{3}\to \mathrm{x}=-60°+\mathrm{k}.180°;\mathrm{k}\in \mathbb{Z}$$

Commented by weltr last updated on 07/Sep/20

$$thanks$$

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