Inverse Trigonometric Functions Set-1

Q1. If $\theta=\tan^{-1}\left(\dfrac{3}{4}\right)$, evaluate $\sin(2\theta)$.

Q2. Solve the equation $\tan^{-1}(x)+\tan^{-1}(2x)+\tan^{-1}(3x)=\pi$ for real $x$.

Q3. Evaluate $\displaystyle \cos\!\left(\sin^{-1}\!\left(\dfrac{3}{5}\right)-\tan^{-1}\!\left(\dfrac{1}{2}\right)\right)$.

Q4. Find the set of real $x$ for which the identity $\displaystyle \tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)=2\tan^{-1}(x)$ holds.

Q5. Evaluate the limit $\displaystyle \lim_{x\to 0}\dfrac{\sin^{-1}x-\tan^{-1}x}{x^3}$.

Q6. If $\theta=\arctan\!\left(\dfrac{3}{4}\right)$, then $\sin(2\theta)=$

Q7. Find all real $x$ satisfying $2\arctan x=\arctan\!\left(\dfrac{2x}{1-x^{2}}\right)$.

Q8. The range of the function $f(x)=\arcsin\!\left(\dfrac{2x}{1+x^{2}}\right)+\arccos\!\left(\dfrac{1-x^{2}}{1+x^{2}}\right)$ for $x\in\mathbb{R}$ is

Q9. Find all real $x$ such that $\arctan x+\arctan(2x)=\dfrac{\pi}{2}$.

Q10. How many real solutions does the equation $\arctan x+\arctan(2x)+\arctan(3x)=\pi$ have?