### Home > PC3 > Chapter 10 > Lesson 10.1.4 > Problem10-83

10-83.

Verify the following trigonometric identities.

1. $\sec(x)-\cos(x)=$$\tan(x)\sin(x)$

$\frac{1}{\cos(x)}-\cos(x)=$

$\frac{1}{\cos(x)}-\frac{\cos^2(x)}{\cos(x)}=$

1. $\sec(x)+\tan(x)=$$\frac{1}{\sec(x)-\tan(x)}$

In this case it is easier to work on the right side of the equation.

$=\frac{1}{\frac{1}{\cos(x)}-\frac{\sin(x)}{\cos(x)}}$

$=\frac{1}{\frac{1-\sin(x)}{\cos(x)}}$

$=\frac{\cos(x)}{1-\sin(x)}$

$=\frac{\cos(x)}{1-\sin(x)}\cdot\frac{1+\sin(x)}{1+\sin(x)}$