Find the principal value of tan^{-1}(2cos(\dfrac{2\pi}{3}))

But we know that  \(cos \dfrac{\pi}{3} = \dfrac{1}{2}\)

So, \( cos \dfrac{2\pi}{3}=\dfrac{-1}{2}\)

Therefore

 tan^-1 (2 × – \( \dfrac{1}{2}\))

= tan^-1(-1)

= \( \dfrac{-\pi}{4}\)

Hence, the principal value of \( tan^{-1 }(2 cos (\dfrac{2\pi}{3}))\) is \( \dfrac{-\pi}{4}\)