Differentiation of Inverse Trigonometric Functions

Q. The value of ${\cot }^{-1}(-1)$ is

1. $\frac{5\pi }{4}$
2. $\frac{3\pi }{4}$
3. $\frac{-\pi }{4}$
4. $\frac{\pi }{4}$

Q. If $y={\cot }^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right](0<x<\pi /2)$ then $\frac{dy}{dx}=$

1. $\frac{1}{2}$
2. $\frac{2}{3}$
3. $3$
4. $1$

Q. If f(x) = x + tan x and f is inverse of g, then g’(x) is equal to

1. $\frac{1}{1+\left(g\right(x)-x{)}^2}$
2. $\frac{1}{1-\left(g\right(x)-x{)}^2}$
3. $\frac{1}{2+\left(g\right(x)-x{)}^2}$
4. $\frac{1}{2-\left(g\right(x)-x{)}^2}$

Q. $Ify=co{s}^{-1}\frac{1-x^2}{1+x^2}then\frac{dy}{dx}atx=-2willbe$

1. $\frac{2}{5}$
2. $-\frac{2}{5}$
3. Does not exist
4. None of the above