Inverse Trigonometric Functions — NDA Mathematics PYQs

Consider the following in respect of inverse circular functions :
I. ${\sin }^{-1}(-x)=-{\sin }^{-1}x$
II. ${\cos }^{-1}(-x)={\cos }^{-1}x$
III. ${\tan }^{-1}(-x)=\pi -{\tan }^{-1}x$
IV. ${\cot }^{-1}(-x)=\pi -{\cot }^{-1}x$
How many of the above are correct ?

What is $\tan \left[2{\tan }^{-1}\frac{1}{2}-\frac{\pi }{4}\right]$ equal to ?

If $k$ is a root of $x^2-4x+1=0$, then what is ${\tan }^{-1}k+{\tan }^{-1}\left(\frac{1}{k}\right)$ equal to?

If ${\tan }^{-1}k+{\tan }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{4}$, then what is the value of $k$?

Consider the following for the two (02) items that follow :
Let $y={\sin }^{-1}\left(x-\frac{4x^3}{27}\right)$.

What is $y$ equal to?

What is ${\cot }^{-1}9+{\text{cosec}}^{-1}\left(\frac{\sqrt{41}}{4}\right)$ equal to?

What is the value of $\tan \left[\frac{1}{2}{\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)\right]$?

If ${\cos }^{-1}x={\sin }^{-1}x$, then which one of the following is correct?

What is $\sqrt{15+{\cot }^2\left(\frac{\pi }{4}-2{\cot }^{-1}3\right)}$ equal to?

What is ${\tan }^{-1}\left(\frac{a}{b}\right)-{\tan }^{-1}\left(\frac{a-b}{a+b}\right)$ equal to?