The simplified expression is the one in option B.
How to simplify the expression?
Remember that:
[tex]sec(x) = 1/cos(x)\\\\cos(x) = cos(-x)[/tex]
With these two, we can rewrite the numerator as:
[tex]sec(x)*sin(-x) + tan(-x)\\\\\frac{1}{cos(x)}*sin(-x) + tan(x)\\\\\frac{1}{cos(-x)}*sin(-x) + tan(-x)\\\\2*tan(-x)[/tex]
Replacing that in the given expression, we have:
[tex]\frac{2*tan(-x)}{1 + sec(-x)} = \frac{2*tan(-x)}{1 + 1/cos(-x)}[/tex]
Now, if we multiply and divide by cos(-x), we get:
[tex]\frac{2*tan(-x)}{1 + 1/cos(-x)} = \frac{2*tan(-x)}{1 + 1/cos(-x)}*\frac{cos(-x)}{cos(-x)} \\\\= \frac{2*sin(-x)}{1 + cos(-x)}[/tex]
Now remember that:
[tex]cos(x) = cos(-x)\\sin(-x) = -sin(x)[/tex]
With these two properties:
[tex]\frac{2*sin(-x)}{1 + cos(-x)} = \frac{-2*sin(x)}{1 + cos(x)}[/tex]
Then the correct option is B.
If you want to learn more about trigonometric identities:
https://brainly.com/question/7331447
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