Certainly! Let's simplify the expression using the double-angle formula for tangent. The expression in question is:
[tex]\[
\frac{2 \tan 25^{\circ}}{1 - \tan^2 25^{\circ}}
\][/tex]
We can recognize this expression as the tan(2θ) formula from trigonometry, where:
[tex]\[
\tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}
\][/tex]
In our case, [tex]\(\theta = 25^{\circ}\)[/tex]. Substituting 25° into the formula, we get:
[tex]\[
\tan(2 \times 25^{\circ}) = \frac{2 \tan 25^{\circ}}{1 - \tan^2 25^{\circ}}
\][/tex]
Therefore, the given expression simplifies directly to:
[tex]\[
\tan(50^{\circ})
\][/tex]
To determine the numerical value of [tex]\(\tan(50^{\circ})\)[/tex], we find:
[tex]\[
\tan(50^{\circ}) \approx 1.19175359259421
\][/tex]
Therefore, the simplified result of the given expression is approximately:
[tex]\[
1.19175359259421
\][/tex]
