Inverse Trigonometric Functions Set-5

Q1. If $\theta=\tan^{-1}\!\left(\frac{3}{4}\right)$, then the value of $\sin 2\theta$ is:

Q2. Find all real solutions of the equation $\tan^{-1}x+\tan^{-1}(2x)+\tan^{-1}(3x)=\pi$.

Q3. The positive real solution of $\sin^{-1}(x)+\tan^{-1}(x)=\dfrac{\pi}{2}$ is:

Q4. Statement A and Statement R are given below.
A: For $x\in(-1,1)$, $\displaystyle \tan^{-1}\!\left(\sqrt{\frac{1-x}{1+x}}\right)=\frac{1}{2}\cos^{-1}x$.
R: Putting $x=\cos\theta$ with $\theta\in(0,\pi)$ gives $\displaystyle \sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=\tan\frac{\theta}{2}$, hence $\tan^{-1}\!\big(\sqrt{\cdots}\big)=\frac{\theta}{2}=\frac{1}{2}\cos^{-1}x$.

Q5. Let $\alpha\in\left(\dfrac{2\pi}{3},\pi\right)$. The principal value of $T=\tan^{-1}\!\left(\dfrac{\sin 3\alpha}{1+\cos 3\alpha}\right)$ equals:

Q6. If $x=\tan^{-1}\!\left(\dfrac{3}{4}\right)$, then $\sin(2x)=$

Q7. Find all real $x$ satisfying

Q8. Let $x\in[-1,1]$. Solve

Q9. Solve for real $x$:

Q10. Find all real $x$ such that

Q11. If $x\in(0,1)$, evaluate $\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$ in terms of $\tan^{-1}x$.

Q12. If $\tan^{-1}x + \tan^{-1}(2x) = \dfrac{\pi}{4}$, then $x=$

Q13. Find the set of real numbers $x$ for which $\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)=2\tan^{-1}x$ holds.

Q14. Assertion (A): For every real $x\neq -1$, $\displaystyle \tan^{-1}x + \tan^{-1}\!\left(\dfrac{1-x}{1+x}\right) = \dfrac{\pi}{4}.$
Reason (R): For every real $x\neq -1$, $\displaystyle \tan\!\Big(\tan^{-1}x + \tan^{-1}\!\Big(\dfrac{1-x}{1+x}\Big)\Big) = 1.$

Q15. How many real solutions does the equation $\sin^{-1}(\sin x)=\dfrac{x}{3}$ have in the interval $[-3\pi,\,3\pi]$?