To determine the correct expression for [tex]\(\tan(2x)\)[/tex], we need to use a well-known trigonometric identity for the tangent of a double angle. The double angle identity for tangent is derived from the sum formula for tangent, [tex]\(\tan(a + b)\)[/tex].
The identity states:
[tex]\[
\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}
\][/tex]
Let’s carefully verify the options provided:
- Option A: [tex]\(\frac{\tan(x)}{1 - \tan^2(x)}\)[/tex]
- Option B: [tex]\(\frac{2 \tan(x)}{1 - \tan^2(x)}\)[/tex]
- Option C: [tex]\(\frac{\tan(x)}{1 + \tan^2(x)}\)[/tex]
- Option D: [tex]\(\frac{2 \tan(x)}{1 + \tan^2(x)}\)[/tex]
According to the correct double angle formula for tangent mentioned above:
[tex]\[
\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}
\][/tex]
We can see that Option B matches the correct identity:
[tex]\[
\frac{2 \tan(x)}{1 - \tan^2(x)}
\][/tex]
Thus, the correct answer is:
B. [tex]\(\frac{2 \tan x}{1-\tan^2 x}\)[/tex]
