$sin\left(2{tan}^{-1}\frac{5}{12}\right)\text{is equal to}$

The correct option is 120/169.

To find the value of $sin\left(2{tan}^{-1}\frac{5}{12}\right)$, we can use trigonometric identity transformations.

Let $\theta ={tan}^{-1}\left(\frac{5}{12}\right)$.
This implies that:
$tan\left(\theta \right)=\frac{5}{12}$

We want to find the value of $sin\left(2\theta \right)$.

Using the double-angle formula for sine in terms of tangent:
$sin\left(2\theta \right)=\frac{2tan\left(\theta \right)}{1+{tan}^2\left(\theta \right)}$

Now, substitute the value of $tan\left(\theta \right)=\frac{5}{12}$ into the formula:

$sin\left(2\theta \right)=\frac{2·\left(\frac{5}{12}\right)}{1+{\left(\frac{5}{12}\right)}^2}$

Simplify the numerator:
$2·\frac{5}{12}=\frac{5}{6}$

Simplify the denominator:
$1+\frac{25}{144}=\frac{144+25}{144}=\frac{169}{144}$

Now divide the simplified numerator by the simplified denominator:

$sin\left(2\theta \right)=\frac{\frac{5}{6}}{\frac{169}{144}}=\frac{5}{6}·\frac{144}{169}$

Since $144÷6=24$, we have:
$sin\left(2\theta \right)=\frac{5·24}{169}=\frac{120}{169}$

Therefore, $sin\left(2{tan}^{-1}\frac{5}{12}\right)=\frac{120}{169}$.