Question

Write the principle value of ${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}(-\sqrt{3})$.

Answer 1

We know that properties of inverse circular functions, $co{t}^{-1}x=\pi –co{t}^{-1}\left(x\right)$ and $ta{n}^{-1}\left(x\right)+co{t}^{-1}\left(x\right)=\frac{\pi }{2}$

Let $x=\sqrt{3}$

$ta{n}^{-1}\left(\sqrt{3}\right)=\frac{\pi }{3}$

$co{t}^{-1}\left(\sqrt{3}\right)=\pi -\frac{\pi }{6}$

So, $ta{n}^{-1}\sqrt{3}-co{t}^{-1}(-\sqrt{3})=\frac{\pi }{3}-(\pi -\frac{\pi }{6})=\frac{-\pi }{2}.$