By using the rules:
[tex]cos(-x) = cos(x)\\\\sec(x) = \frac{1}{cos(x)}[/tex]
We have proven the identity. Below you can follow the demonstration.
How to prove the identity?
Here you need to remember two things:
[tex]cos(-x) = cos(x)\\\\sec(x) = \frac{1}{cos(x)}[/tex]
Here we have the expression:
[tex]\frac{1 - cos(-x)}{sec(-x) - 1}[/tex]
By using the first rule, we can rewrite:
[tex]\frac{1 - cos(-x)}{sec(-x) - 1} = \frac{1 - cos(x)}{sec(x) - 1}[/tex]
By using the second rule, we can rewrite:
[tex]\frac{1 - cos(x)}{sec(x) - 1} = \frac{1 - cos(x)}{\frac{1}{cos(x)} - 1}[/tex]
Now if we multiply and divide by cos(x), we get:
[tex]\frac{1 - cos(x)}{\frac{1}{cos(x)} - 1} = \frac{1 - cos(x)}{\frac{1}{cos(x)} - 1} *\frac{cos(x)}{cos(x) } = \frac{(1- cos(x))*cos(x)}{1 - cos(x)} = cos(x)[/tex]
In this way, the identity was proven.
If you want to learn more about trigonometric identities:
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