Differentiation of Inverse Trigonometric Functions

Q. If $tan\theta +tan2\theta +tan3\theta =0then\theta =$

1. 
2. 
3. 
4. 

Q.

Prove the following:
cos 6x = $32{\text{cos}}^{6}x-48{\text{cos}}^{4}x+18{\text{cos}}^{2}x-1$

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. If $y=ta{n}^{-1}\frac{1}{1+x+x^2}+ta{n}^{-1}\frac{1}{x^2+3x+3}+ta{n}^{-1}\frac{1}{x^2+5x+7}+\cdots +$ upto n terms, then $y^{\prime }\left(0\right)$ is equal to

1. $-\frac{1}{1+n^2}$
2. $-\frac{n^2}{1+n^2}$
3. $\frac{n}{1+n^2}$
4. None of these

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. $1$