# Complex Modulus/Examples/Tangent of Angle + i

## Example of Complex Modulus

$\left\vert{\tan \theta + i}\right\vert = \left\vert{\sec \theta}\right\vert$

where:

$\theta \in \R$ is a real number
$\tan \theta$ denotes the tangent function
$\sec \theta$ denotes the secant function.

## Proof

 $\ds \left\vert{\tan \theta + i}\right\vert$ $=$ $\ds \left\vert{\tan \theta + 1 i}\right\vert$ $\ds$ $=$ $\ds \sqrt {\tan^2 \theta + 1}$ Definition of Complex Modulus $\ds$ $=$ $\ds \sqrt {\sec^2 \theta}$ $\ds$ $=$ $\ds \left\vert{\sec \theta}\right\vert$ Absolute Value Equals Square Root of Square

$\blacksquare$