Inverse Trigonometric Functions Set-6

Q1. Let $\theta=\sin^{-1}\!\left(\frac{3}{5}\right)$ and $\phi=\cos^{-1}\!\left(\frac{5}{13}\right)$. Find $\sin(\theta+\phi)$.

Q2. Evaluate $\tan^{-1}2+\tan^{-1}3-\tan^{-1}1.$

Q3. Find all real $x$ satisfying $\sin^{-1}\!\left(\frac{2x}{1+x^2}\right)=2\tan^{-1}x.$

Q4. Find the real pair $(x,y)$ satisfying the system

Q5. Evaluate $\tan^{-1}\!\left(\frac{1}{2}\right)+\tan^{-1}\!\left(\frac{1}{5}\right)+\tan^{-1}\!\left(\frac{1}{8}\right).$

Q6. Solve for real $x$: $\sin^{-1}\!\left(\frac{2x}{1+x^{2}}\right)=2\tan^{-1}x$

Q7. Evaluate $\tan^{-1}\!\frac{1}{2}+\tan^{-1}\!\frac{1}{5}+\tan^{-1}\!\frac{1}{8}$

Q8. Find the real $x$ (with $|x|\le1$) satisfying $\sin^{-1}x+\sin^{-1}(2x)=\frac{\pi}{2}$

Q9. Solve for real $x$: $\sin^{-1}\!\left(\frac{2x}{1+x^{2}}\right)=\cos^{-1}\!\left(\frac{1-x^{2}}{1+x^{2}}\right)$

Q10. If $-\dfrac{\pi}{2}<\theta<\dfrac{\pi}{2}$, simplify: $\tan^{-1}\!\left(\sqrt{\frac{1+\sin\theta}{1-\sin\theta}}\right)$

Q11. (If real numbers $a,b$ satisfy $ab=\dfrac{1}{3}$ and $\arctan a+\arctan b=\dfrac{\pi}{4}$, then $a+b=$ )

Q12. (Find all real solutions of $\arctan(2x)+\arctan(3x)=\dfrac{\pi}{4}$.)

Q13. (For which real $x$ is the equality $\arctan x+\arctan\!\left(\dfrac{1-x}{1+x}\right)=\dfrac{\pi}{4}$ valid?)

Q14. (Solve for real $x$ (if any) satisfying $\arcsin x+\arctan x=\dfrac{\pi}{2}$.)

Q15. (Solve for real $x$ such that $\arctan x+\arctan(2x)+\arctan(3x)=\pi$.)