The values of $tan^2(sec^{-1}2)+cot^2(cosec^{-1}3),$ is

11

The correct answer is Option (3) → 11

$\theta = \sec^{-1}(2) \Rightarrow \sec\theta = 2$

$\cos\theta = \frac{1}{2}, \quad \sin\theta = \frac{\sqrt{3}}{2}$

$\tan\theta = \sqrt{3} \Rightarrow \tan^2(\sec^{-1}2) = 3$

$\phi = \sin^{-1}\left(\frac{1}{3}\right)$

$\sin\phi = \frac{1}{3}, \quad \cos\phi = \frac{2\sqrt{2}}{3}$

$\cot\phi = \frac{\cos\phi}{\sin\phi} = 2\sqrt{2}$

$\cot^2(\sin^{-1}\frac{1}{3}) = 8$

$\text{Sum} = 3 + 8 = 11$

The value is $11$.