Inverse Trigonometric Fun question bank (Maths)

Let $\alpha = 3\sin^{-1}\left(\dfrac{6}{11}\right)$ and $\beta = 3\cos^{-1}\left(\dfrac{4}{9}\right)$, where inverse trigonometric functions take only the principal values.
Given below are two statements:
Statement I: $\cos(\alpha+\beta) \gt 0$.
Statement II: $\cos(\alpha) \lt 0$.
In the light of the above statements, choose the correct answer from the options given below:

If $\sin(\tan^{-1}(x\sqrt{2})) = \cot(\sin^{-1}\sqrt{1-x^2})$, $x \in (0,1)$, then the value of $x$ is :