1.Find the domain of the function $f(x) = \sin^{-1}\left(\frac{x^2+1}{2x}\right)$.
2.Find the exact principal value of $\cos^{-1}(\cos 10)$ in radians.
3.Find the exact principal value of $\sin^{-1}(\sin 10)$ in radians.
4.Determine the domain of the real-valued function $f(x) = \sqrt{\sin^{-1}(2x) + \frac{\pi}{6}}$.
5.Find the range of the function $f(x) = 2\sin^{-1}x + \cos^{-1}x$.
6.Simplify the expression: $\tan^{-1}\left(\frac{\sqrt{1+x^2} - \sqrt{1-x^2}}{\sqrt{1+x^2} + \sqrt{1-x^2}}\right)$.
7.Evaluate: $\tan\left(\frac{1}{2}\sin^{-1}\frac{2x}{1+x^2} + \frac{1}{2}\cos^{-1}\frac{1-y^2}{1+y^2}\right)$, given that $x > 1, y > 1$, and $xy > 1$.
8.Prove that: $\cot^{-1}\left(\frac{ab+1}{a-b}\right) + \cot^{-1}\left(\frac{bc+1}{b-c}\right) + \cot^{-1}\left(\frac{ca+1}{c-a}\right) = 0$, given $a > b > c > 0$.
9.Find the value of the series sum: $\sum_{r=1}^{n} \tan^{-1}\left(\frac{1}{1 + r + r^2}\right)$.
10.Find the infinite sum: $\sum_{n=1}^{\infty} \tan^{-1}\left(\frac{2}{n^2}\right)$.
11.If $\sin^{-1}a + \sin^{-1}b + \sin^{-1}c = \frac{3\pi}{2}$, find the value of $\frac{a^{2026} + b^{2026} + c^{2026}}{a^{2026} b^{2026} c^{2026}}$.
12.If $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi$, prove that $x^2 + y^2 + z^2 + 2xyz = 1$.
13.Evaluate the exact numerical value of: $\cos\left(\frac{1}{2}\cos^{-1}\left(\frac{1}{8}\right)\right)$.
14.Simplify: $\sin^{-1}\left(\frac{5}{13}\cos x + \frac{12}{13}\sin x\right)$.
15.Find the maximum and minimum values of the function $f(x) = (\sin^{-1}x)^2 + (\cos^{-1}x)^2$.
16.Solve the inequality: $\sin^{-1}x > \cos^{-1}x$.
17.Solve for $x$: $\sin^{-1}(6x) + \sin^{-1}(6\sqrt{3}x) = -\frac{\pi}{2}$.
18.Solve for $x$: $\cos^{-1}x - \sin^{-1}x = \cos^{-1}(x\sqrt{3})$.
19.Find the number of real solutions of the equation: $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \frac{\pi}{2}$.
20.If $\tan^{-1}x + \tan^{-1}y = \frac{4\pi}{5}$, find the exact value of $\cot^{-1}x + \cot^{-1}y$.
21.Solve for $x$: $\sec^{-1}\left(\frac{x}{a}\right) - \sec^{-1}\left(\frac{x}{b}\right) = \sec^{-1}b - \sec^{-1}a$, where $a, b > 0, a \neq b$.
22.Solve the equation: $\sin^{-1}\left(\frac{1}{\sqrt{5}}\right) + \cos^{-1}x = \frac{\pi}{4}$.
23.If $y = \cot^{-1}(\sqrt{\cos x}) - \tan^{-1}(\sqrt{\cos x})$, prove that $\sin y = \tan^2\left(\frac{x}{2}\right)$.
24.Solve for $x$: $\tan^{-1}(x-1) + \tan^{-1}x + \tan^{-1}(x+1) = \tan^{-1}(3x)$.
25.If $x, y, z$ are in Arithmetic Progression (A.P.) and $\tan^{-1}x, \tan^{-1}y, \tan^{-1}z$ are also in A.P., prove that either $x = y = z$ or $y = 0$.
26.Solve for $x$: $\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \frac{4\pi}{3}$.
27.Find the value of $x$ satisfying: $\tan^{-1}\left(\frac{1}{4}\right) + 2\tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{6}\right) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{4}$.
28.Find the domain of $f(x) = \sec^{-1}\left(x - \frac{1}{x}\right)$.
29.If $\cos^{-1}\frac{x}{a} + \cos^{-1}\frac{y}{b} = \alpha$, prove that $\frac{x^2}{a^2} - \frac{2xy}{ab}\cos\alpha + \frac{y^2}{b^2} = \sin^2\alpha$.
30.Evaluate: $\sin\left(\cot^{-1}\left(\cos\left(\tan^{-1}x\right)\right)\right)$.
