Consider the following two statements P and Q. P: ^ -1 (cos...

We have, $\cos \left(\dfrac{4\pi}{3}\right) = - cos \dfrac{\pi}{3} = - \dfrac{1}{2}$

Hence, $\cos^{- 1} \left(- \dfrac{1}{2}\right) = \dfrac{2\pi}{3}$
Therefore, $P$ is false.

Now, $\sec^{2} \left(\cot^{- 1} \dfrac{1}{2}\right) + cose{c}^{2}\left(\tan^{- 1} \dfrac{1}{3}\right)$

$= \left[1 + \tan^{2} \left(\tan^{- 1} 2\right)\right] + \left[1 + \cot^{2} \left(\cot^{- 1} 3\right)\right]$
$= 1 + 4 + 1 + 9 = 15$
Therefore, $Q$ is true.