The value of $\tan^2 (\sec^{-1}2)+ \cot^2(cosec^{-1}3)$ is equal to

11

The correct answer is Option (2) → 11

Let $\theta = \sec^{-1}2$ ⇒ $\sec\theta = 2$ ⇒ $\cos\theta = \frac{1}{2}$

Then $\tan^{2}\theta = \sec^{2}\theta - 1 = 4 - 1 = 3$

Let $\phi = \csc^{-1}3$ ⇒ $\csc\phi = 3$ ⇒ $\sin\phi = \frac{1}{3}$

Then $\cot^{2}\phi = \csc^{2}\phi - 1 = 9 - 1 = 8$

Hence, $\tan^{2}(\sec^{-1}2) + \cot^{2}(\csc^{-1}3) = 3 + 8 = 11$