# How do you verify the identify tan^2theta+1=sec^2theta?

Aug 3, 2016

see explanation

#### Explanation:

The following $\textcolor{b l u e}{\text{trigonometric identies}}$ are 'useful'

$\textcolor{\mathmr{and} a n \ge}{\text{Reminders}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\tan \theta = \frac{\sin \theta}{\cos \theta}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\sin}^{2} \theta + {\cos}^{2} \theta = 1} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ and } \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\sec \theta = \frac{1}{\cos} \theta} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

the left side $= {\tan}^{2} \theta + 1 = \frac{{\sin}^{2} \theta}{{\cos}^{2} \theta} + \frac{{\cos}^{2} \theta}{{\cos}^{2} \theta}$

Expressing as a single fraction gives.

$\frac{{\sin}^{2} \theta + {\cos}^{2} \theta}{{\cos}^{2} \theta} = \frac{1}{{\cos}^{2} \theta} = {\sec}^{2} \theta = \text{right side hence verified}$