1.Find the principal value of $\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)$.
2.Find the principal value of $\cos^{-1}\left(\frac{1}{2}\right)$.
3.Find the principal value of $\tan^{-1}(\sqrt{3})$.
4.Find the principal value of $\text{cosec}^{-1}(2)$.
5.Find the principal value of $\sec^{-1}\left(\frac{2}{\sqrt{3}}\right)$.
6.Find the principal value of $\cot^{-1}(\sqrt{3})$.
7.Find the principal value of $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$.
8.Find the principal value of $\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)$.
9.Find the principal value of $\tan^{-1}(-1)$.
10.Find the principal value of $\text{cosec}^{-1}(-\sqrt{2})$.
11.Evaluate: $\sin(\sin^{-1} 0.5)$.
12.Evaluate: $\cos(\cos^{-1} \frac{\sqrt{3}}{2})$.
13.Evaluate: $\sin^{-1}\left(\sin \frac{\pi}{4}\right)$.
14.Evaluate: $\cos^{-1}\left(\cos \frac{2\pi}{3}\right)$.
15.Evaluate: $\sin^{-1}\left(\sin \frac{2\pi}{3}\right)$. (Hint: Check domain constraint)
16.Evaluate: $\cos^{-1}\left(\cos \frac{7\pi}{6}\right)$. (Hint: Check domain constraint)
17.Evaluate: $\tan^{-1}\left(\tan \frac{3\pi}{4}\right)$.
18.Write $\text{cosec}^{-1}(x)$ in terms of $\sin^{-1}$ and use it to find the value of $\text{cosec}^{-1}(2)$.
19.Evaluate $\sec^{-1}(-2)$ using the reciprocal property.
20.Evaluate: $\sin\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right)$.
21.Find the value of $\sin^{-1}(-x) + \sin^{-1}(x)$.
22.Find the value of $\cos^{-1}\left(-\frac{1}{2}\right) + \cos^{-1}\left(\frac{1}{2}\right)$.
23.Find the value of $\sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}\left(\frac{1}{5}\right)$.
24.Find the value of $\tan^{-1}\left(\frac{2}{3}\right) + \cot^{-1}\left(\frac{2}{3}\right)$.
25.Find the value of $\sin\left(\sin^{-1}\frac{1}{2} + \cos^{-1}\frac{1}{2}\right)$.
26.Find the value of $\cos(\sec^{-1}x + \text{cosec}^{-1}x)$, where $|x| \ge 1$.
27.Simplify: $\cos^{-1}(-x) + \cos^{-1}(x)$.
28.If $\sin^{-1}x = \frac{\pi}{5}$, find the value of $\cos^{-1}x$.
29.If $\tan^{-1}x = \frac{\pi}{4}$, find the value of $\cot^{-1}x$.
30.Evaluate: $\cot(\tan^{-1}a + \cot^{-1}a)$.
31.Evaluate: $\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right)$.
32.Evaluate: $\tan^{-1}(2) + \tan^{-1}(3)$. (Careful: $xy > 1$)
33.Evaluate: $\tan^{-1}\left(\frac{2}{11}\right) + \tan^{-1}\left(\frac{7}{24}\right)$.
34.Express $2\tan^{-1}\left(\frac{1}{3}\right)$ as a single $\tan^{-1}$ function.
35.Express $2\tan^{-1}\left(\frac{1}{2}\right)$ as a single $\sin^{-1}$ function.
36.Express $2\tan^{-1}\left(\frac{1}{3}\right)$ as a single $\cos^{-1}$ function.
37.Evaluate: $\tan\left(\tan^{-1}\frac{1}{4} + \tan^{-1}\frac{2}{9}\right)$.
38.Simplify: $\tan^{-1}\left(\frac{x}{y}\right) - \tan^{-1}\left(\frac{x-y}{x+y}\right)$.
39.Find the value of $\tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3)$.
40.If $\tan^{-1}x + \tan^{-1}y = \frac{\pi}{4}$ and $xy < 1$, write the equation relating $x$ and $y$.
41.Simplify: $\tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right)$, where $-\pi < x < \pi$.
42.Simplify: $\tan^{-1}\left(\frac{\cos x}{1 - \sin x}\right)$.
43.Simplify: $\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$, where $0 < x < \pi$.
44.Simplify: $\cot^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right)$, for $x > 1$.
45.Simplify: $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$.
46.Simplify: $\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$.
47.Simplify: $\tan^{-1}\left(\frac{3a^2x - x^3}{a^3 - 3ax^2}\right)$, where $a > 0$.
48.Simplify: $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$, where $-\frac{1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}}$.
49.Simplify: $\cos^{-1}(4x^3 - 3x)$, where $\frac{1}{2} \le x \le 1$.
50.Simplify: $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$, for $x \neq 0$.
51.Solve for $x$: $\sin^{-1}x = \frac{\pi}{6}$.
52.Solve for $x$: $\tan^{-1}x = -\frac{\pi}{4}$.
53.Solve for $x$: $\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$.
54.Solve for $x$: $\sin^{-1}(1-x) - 2\sin^{-1}x = \frac{\pi}{2}$.
55.Solve for $x$: $\tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4}$.
56.Solve for $x$: $2\tan^{-1}(\cos x) = \tan^{-1}(2\text{cosec } x)$.
57.Solve for $x$: $\cos(\tan^{-1}x) = \sin\left(\cot^{-1}\frac{3}{4}\right)$.
58.Solve for $x$: $\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{1}{2}\tan^{-1}x$, for $x > 0$.
59.Solve for $x$: $\sin^{-1}\left(\frac{5}{13}\right) + \cos^{-1}x = \frac{\pi}{2}$.
60.Solve for $x$: $\sin^{-1}x + \sin^{-1}(2x) = \frac{\pi}{3}$.
