Examining the question:
We are given the expression:
[tex]\frac{1}{Sec(\alpha)-Tan(\alpha)}[/tex]
We know from Basic trigonometry that:
[tex]Sec(\alpha) = \frac{1}{Cos(\alpha)}[/tex]
[tex]Tan(\alpha) = \frac{Sin(\alpha)}{Cos(\alpha)}[/tex]
Simplifying the expression:
Replacing these values in the given expression, we get:
[tex]\frac{1}{\frac{1}{Cos(\alpha)} -\frac{Sin(\alpha)}{Cos(\alpha)} }[/tex]
Since the denominator of both the values in the denominator is the same:
[tex]\frac{1}{\frac{1-Sin(\alpha)}{Cos(\alpha)} }[/tex]
We know that [tex]\frac{1}{\frac{a}{b} }[/tex] = [tex]\frac{b}{a}[/tex], using the same property:
[tex]\frac{Cos(\alpha)}{1-Sin(\alpha)}[/tex]
and we are done!
