# How do you verify the identity sec2theta=(cos^2theta+sin^2theta)/(cos^2theta-sin^2theta)?

Nov 20, 2016

The equation is verified by applying some trigonometric identities.
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Trigonometric identities used in this exercise:
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$\textcolor{red}{{\cos}^{2} x + {\sin}^{2} x = 1} \text{ } \textcolor{red}{E q 1}$
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$\textcolor{b l u e}{{\cos}^{2} x - {\sin}^{2} x = \cos 2 x} \text{ } \textcolor{b l u e}{E q 2}$
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$\sec x = \frac{1}{\cos} x \text{ } E q 3$
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Taking the second side of the equation to prove the equality.
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$\frac{{\cos}^{2} \theta + {\sin}^{2} \theta}{{\cos}^{2} \theta - {\sin}^{2} \theta}$
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$= \frac{\textcolor{red}{1}}{{\cos}^{2} \theta - {\sin}^{2} \theta} \text{ }$applying $\textcolor{red}{E q 1}$

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$= \frac{\textcolor{red}{1}}{\textcolor{b l u e}{\cos 2 \theta}} \text{ }$applying $\textcolor{b l u e}{E q 2}$
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$= \sec 2 \theta \text{ }$ Applying $E q 3 \text{ }$ for $x = 2 \theta$