Question

$\tan\left(\tan^{-1}(-1)+\frac{\pi}{3}\right)$

Answer 1

2 - $\sqrt{3}$

Answer 2

Evaluate $\tan^{-1}(-1)$ to find $-\frac{\pi}{4}$. Substitute this into the expression to get $\tan(-\frac{\pi}{4} + \frac{\pi}{3})$. Apply the tangent addition formula $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ with $\tan(-\frac{\pi}{4}) = -1$ and $\tan(\frac{\pi}{3}) = \sqrt{3}$. This yields $\frac{-1+\sqrt{3}}{1-(-1)\sqrt{3}} = \frac{\sqrt{3}-1}{1+\sqrt{3}}$. Rationalizing the denominator gives the final answer $2-\sqrt{3}$.