[tex]\frac{\csc (x)-\sec (x)}{\csc (x)+\sec (x)}[/tex]
Let's rewrite the expression as:
[tex]\frac{\frac{1}{\sin(x)}-\frac{1}{\cos(x)}}{\frac{1}{\sin(x)}+\frac{1}{\cos (x)}}[/tex]
Multiply the numerator and the denominator by sin(x):
[tex]\begin{gathered} \frac{\frac{\sin(x)}{\sin(x)}-\frac{\sin(x)}{\cos(x)}}{\frac{\sin(x)}{\sin(x)}+\frac{\sin (x)}{\cos (x)}} \\ _{\text{ }}where\colon \\ \frac{\sin(x)}{\cos(x)}=\tan (x) \\ so\colon \\ \frac{1-\tan (x)}{1+\tan (x)} \end{gathered}[/tex]
