Why am I getting a different value for $\sin\left(2\tan^{-1}\frac{4}{3}\right)$ than my calculator? The expression:
$$\sin\left(2\tan^{-1}\left(\frac{4}{3}\right)\right)$$
Way 1:
If I punch the above expression in my calculator, I get $\frac{24}{25}$.
Way 2:
$$\sin\left(2\tan^{-1}\left(\frac{4}{3}\right)\right)$$
$$\sin\left(\tan^{-1}\left(\frac{2\times\frac{4}{3}}{1-(\frac{4}{3})^{2}}\right)\right)$$
$$[\text{Using the formula $2\arctan(x)=\arctan\left(\frac{2x}{1-x^2}\right)$}]$$
$$\sin\left(\tan^{-1}\left(\frac{-24}{7}\right)\right)$$
$$-\sin\left(\tan^{-1}\left(\frac{24}{7}\right)\right)$$
$$-\sin\left(\sin^{-1}\left(\frac{24}{25}\right)\right)$$
$$-\frac{24}{25}$$
Why am I getting a different answer than that of my calculator?

Notice that your formula to simplify the arctan gives a positive answer on the left hand side
$$2\arctan \frac{4}{3} > 0$$
but a negative on the right
$$\arctan \frac{\frac{8}{3}}{1-\left(\frac{4}{3}\right)^2} < 0$$
The formula itself is the mistake.
