Which of the following is the principal value branch of \(\cos^{-1} x\)?
\([-\dfrac{\pi}{2}, \dfrac{\pi}{2}]\)
(0, \pi)
[0, \pi]
(0, \dfrac{\pi}{2}]
Which of the following is the principal value branch of \(\csc^{-1} x\)?
\((-\dfrac{\pi}{2}, -\dfrac{\pi}{2})\)
[0, \pi] - \{\dfrac{\pi}{2}\}
\([-\dfrac{\pi}{2}, \dfrac{\pi}{2}]\)
\([-\dfrac{\pi}{2}, \dfrac{\pi}{2}] - \{0\}\)
If \(3 \tan^{-1} x + \cot^{-1} x = \pi\), then \(x\) equals
0
1
-1
\(\dfrac{1}{2}\)
The value of \(\sin^{-1}\left(\cos \dfrac{33\pi}{5}\right)\) is
\(\dfrac{3\pi}{5}\)
\(-\dfrac{7\pi}{5}\)
\(\dfrac{\pi}{10}\)
\(-\dfrac{\pi}{10}\)
The domain of the function \(\cos^{-1}(2x - 1)\) is
[0, 1]
[-1, 1]
(-1, 1)
[0, \pi]
The domain of the function defined by \(f(x) = \sin^{-1} \sqrt{x - 1}\) is
[1, 2]
[-1, 1]
[0, 1]
none of these
If \(\cos\left(\sin^{-1} \dfrac{2}{5} + \cos^{-1} x\right) = 0\), then \(x\) is equal to
\(\dfrac{1}{5}\)
\(\dfrac{2}{5}\)
0
1
The value of \(\sin(2 \tan^{-1}(0.75))\) is equal to
.75
1.5
.96
\(\sin^{-1} 5\)
The value of \(\cos^{-1}(\cos \dfrac{3\pi}{2})\) is equal to
\(\dfrac{\pi}{2}\)
\(3\pi\)
\(\dfrac{5\pi}{2}\)
\(7\pi\)
The value of the expression \(2 \sec^{-1} 2 + \sin^{-1} \left(\dfrac{1}{2}\right)\) is
\(\dfrac{\pi}{6}\)
\(\dfrac{5\pi}{6}\)
\(\dfrac{7\pi}{6}\)
1
If \(\tan^{-1} x + \tan^{-1} y = \dfrac{4\pi}{5}\), then \(\cot^{-1} x + \cot^{-1} y\) equals
\(\dfrac{\pi}{5}\)
\(\dfrac{2\pi}{5}\)
\(\dfrac{3\pi}{5}\)
\(\pi\)
If \(\sin^{-1}\left(\dfrac{2a}{1 + a^{2}}\right) + \cos^{-1}\left(\dfrac{1 - a^{2}}{1 + a^{2}}\right) = \tan^{-1}\left(\dfrac{2x}{1 - x^{2}}\right)\), where \(a, x \in (0, 1)\), then the value of \(x\) is
0
\(\dfrac{a}{2}\)
a
\(\dfrac{2a}{1 - a^{2}}\)
The value of \(\cot \left[ \cos^{-1} \left(\dfrac{7}{25}\right) \right]\) is
\(\dfrac{25}{24}\)
\(\dfrac{25}{7}\)
\(\dfrac{24}{25}\)
\(\dfrac{7}{24}\)
The value of the expression \(\tan\left(\dfrac{1}{2} \cos^{-1} \dfrac{2}{\sqrt{5}}\right)\) is
\(2 + \sqrt{5}\)
\(\sqrt{5} - 2\)
\(\dfrac{\sqrt{5} + 2}{2}\)
5 + \sqrt{2}
If \(x \le 1\), then \(2 \tan^{-1} x + \sin^{-1}\left(\dfrac{2x}{1 + x^{2}}\right)\) is equal to
\(4 \tan^{-1} x\)
0
\(\dfrac{\pi}{2}\)
\(\pi\)
If \(\cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi\), then \(\alpha(\beta + \gamma) + \beta(\gamma + \alpha) + \gamma(\alpha + \beta)\) equals
0
1
6
12
The number of real solutions of the equation \(\sqrt{1 + \cos 2x} = \sqrt{2} \cos^{-1}(\cos x)\) in \([\dfrac{\pi}{2}, \pi]\) is
0
1
2
Infinite
If \(\cos^{-1} x > \sin^{-1} x\), then
\(\dfrac{1}{\sqrt{2}} < x \le 1\)
0 \le x < \dfrac{1}{\sqrt{2}}\)
-1 \le x < \dfrac{1}{\sqrt{2}}\)
x > 0