Question

Evaluate : $\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\left(\frac{2\pi}{3}\right)\right)$}

Hint:

Remember that $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$ holds true for identical values of $x$. However, here the inputs are $-\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{3}}$, so evaluate them directly rather than misapplying the identity.

Answer 1