1.Find the principal value of $\sin^{-1}\left(\sin \frac{4\pi}{5}\right)$.
2.Find the principal value of $\cos^{-1}\left(\cos \frac{13\pi}{6}\right)$.
3.Find the principal value of $\tan^{-1}\left(\tan \frac{7\pi}{6}\right)$.
4.Find the principal value of $\sin^{-1}\left(\sin \frac{3\pi}{5}\right)$.
5.Find the value of $\sec^{-1}(-2) + \text{cosec}^{-1}\left(- \frac{2}{\sqrt{3}}\right)$.
6.Evaluate: $\tan^{-1}(-\sqrt{3}) + \sec^{-1}(-2) - \text{cosec}^{-1}\left(\frac{2}{\sqrt{3}}\right)$.
7.Evaluate: $\sin^{-1}\left(-\frac{1}{2}\right) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \cot^{-1}(-\sqrt{3})$.
8.Find the principal value of $\cos^{-1}\left(\cos \frac{5\pi}{4}\right)$.
9.Find the domain of the function $f(x) = \sin^{-1}(2x - 3)$.
10.Find the domain of the function $f(x) = \cos^{-1}(x^2 - 4)$.
11.Evaluate: $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$.
12.Evaluate: $\cos\left(\sin^{-1}\frac{3}{5}\right)$.
13.Evaluate: $\tan\left(\cos^{-1}\frac{8}{17}\right)$.
14.Evaluate: $\sin\left(2 \sin^{-1} 0.6\right)$.
15.Evaluate: $\cos\left(\sec^{-1} \frac{5}{3}\right)$.
16.Express $\sin(\cot^{-1} x)$ in terms of $x$.
17.Write $\tan\left(\text{cosec}^{-1} \frac{\sqrt{5}}{2}\right)$ in its simplest form.
18.Evaluate: $\sin\left(\tan^{-1}(-\sqrt{3}) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$.
19.Evaluate: $\tan\left(\frac{1}{2} \cos^{-1} \frac{\sqrt{5}}{3}\right)$.
20.Find the range of the function $f(x) = \sin^{-1}x + \cos^{-1}x$.
21.If $\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}$, find the value of $\cos^{-1} x + \cos^{-1} y$.
22.Solve for $x$: $\sin\left(\sin^{-1}\frac{1}{5} + \cos^{-1}x\right) = 1$.
23.Solve for $x$: $\cos(\tan^{-1} x) = \sin\left(\cot^{-1} \frac{3}{4}\right)$.
24.Simplify: $\tan^{-1}\left(\frac{a-b}{1+ab}\right) + \tan^{-1}\left(\frac{b-c}{1+bc}\right) + \tan^{-1}\left(\frac{c-a}{1+ca}\right)$.
25.If $4\sin^{-1} x + \cos^{-1} x = \pi$, find the value of $x$.
26.Simplify: $\sin^{-1}x + \sin^{-1}(-x) + \cos^{-1}x + \cos^{-1}(-x)$.
27.Solve for $x$: $\cot\left(\sin^{-1} \frac{5}{13}\right) = \tan(\cos^{-1} x)$.
28.If $\cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi$, find the value of $\alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta)$.
29.Evaluate: $\sin^2\left(\cos^{-1}\frac{1}{2}\right) + \cos^2\left(\sin^{-1}\frac{1}{3}\right)$.
30.Find the maximum possible value of $f(x) = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x$.
31.Evaluate: $\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{8}$.
32.Evaluate: $\tan\left(2 \tan^{-1} \frac{1}{5} - \frac{\pi}{4}\right)$.
33.Evaluate: $\sin\left(2 \tan^{-1} \frac{2}{3}\right) + \cos(\tan^{-1} \sqrt{3})$.
34.Simplify: $\tan^{-1}\left(\frac{\sqrt{x} + \sqrt{y}}{1 - \sqrt{xy}}\right)$.
35.If $\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-y^2}{1+y^2}\right) = 2\tan^{-1} a$, find $a$ in terms of $x$ and $y$.
36.Evaluate: $\cos\left[2\tan^{-1}\left(\frac{1}{2}\right)\right] + \sin\left[2\tan^{-1}\left(\frac{1}{3}\right)\right]$.
37.If $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi$, what is $x+y+z$ equal to?
38.If $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \frac{\pi}{2}$, what is $xy + yz + zx$ equal to?
39.Evaluate: $\tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3)$.
40.Evaluate: $\tan^{-1} \frac{3}{4} + \tan^{-1} \frac{3}{5} - \tan^{-1} \frac{8}{19}$.
41.Simplify: $\tan^{-1}\left(\frac{\sqrt{1+x^2} - 1}{x}\right)$, $x \neq 0$.
42.Simplify: $\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$, where $0 < x < \pi$.
43.Simplify: $\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right)$, where $0 < x < \frac{\pi}{2}$.
44.Simplify: $\sin^{-1}\left(\frac{x}{\sqrt{x^2+a^2}}\right)$.
45.Simplify: $\tan^{-1}\left(\frac{a \cos x - b \sin x}{b \cos x + a \sin x}\right)$.
46.Simplify: $\sec^{-1}\left(\frac{1}{2x^2-1}\right)$, where $0 < x < \frac{1}{\sqrt{2}}$.
47.Simplify: $\sin^{-1}\left(x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right)$.
48.Simplify: $\tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$.
49.Simplify: $\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$.
50.Simplify: $\tan^{-1}\left(\sqrt{\frac{a-x}{a+x}}\right)$, where $-a < x < a$.
51.Solve for $x$: $\sin^{-1} x + \sin^{-1} 2x = \frac{\pi}{3}$.
52.Solve for $x$: $\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}$.
53.Solve for $x$: $\cos^{-1} x + \sin^{-1} \frac{x}{2} = \frac{\pi}{6}$.
54.Solve for $x$: $2\tan^{-1}(\cos x) = \tan^{-1}(2\text{cosec } x)$.
55.Solve for $x$: $\sin^{-1}(1-x) - 2\sin^{-1}x = \frac{\pi}{2}$.
56.Solve for $x$: $\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{1}{2}\tan^{-1}x$, for $x > 0$.
57.Solve for $x$: $\sin^{-1}\frac{5}{x} + \sin^{-1}\frac{12}{x} = \frac{\pi}{2}$.
58.Solve for $x$: $\tan^{-1}(x+1) + \tan^{-1}(x-1) = \tan^{-1}\frac{8}{31}$.
59.Solve for $x$: $3\sin^{-1}\left(\frac{2x}{1+x^2}\right) - 4\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 2\tan^{-1}\left(\frac{2x}{1-x^2}\right) = \frac{\pi}{3}$.
60.Solve for $x$: $\cos^{-1}\left(\frac{x^2-1}{x^2+1}\right) + \tan^{-1}\left(\frac{2x}{x^2-1}\right) = \frac{2\pi}{3}$.
