How do you simplify sqrt(1+tan^2x)?

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Explanation:

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Jan 13, 2017

$\sqrt{1 + {\tan}^{2} x} = \left\mid \sec x \right\mid$

Explanation:

Using:

${\cos}^{2} x + {\sin}^{2} x = 1$

$\tan x = \sin \frac{x}{\cos} x$

$\sec x = \frac{1}{\cos} x$

we find:

$\sqrt{1 + {\tan}^{2} x} = \sqrt{1 + \frac{{\sin}^{2} x}{{\cos}^{2} x}}$

$\textcolor{w h i t e}{\sqrt{1 + {\tan}^{2} x}} = \sqrt{\frac{{\cos}^{2} x}{{\cos}^{2} x} + \frac{{\sin}^{2} x}{{\cos}^{2} x}}$

$\textcolor{w h i t e}{\sqrt{1 + {\tan}^{2} x}} = \sqrt{\frac{{\cos}^{2} x + {\sin}^{2} x}{{\cos}^{2} x}}$

$\textcolor{w h i t e}{\sqrt{1 + {\tan}^{2} x}} = \sqrt{\frac{1}{{\cos}^{2} x}}$

$\textcolor{w h i t e}{\sqrt{1 + {\tan}^{2} x}} = \sqrt{{\sec}^{2} x}$

$\textcolor{w h i t e}{\sqrt{1 + {\tan}^{2} x}} = \left\mid \sec x \right\mid$

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