Find the value of \( \tan^{-1}(\tan 5\pi/6) + \cos^{-1}(\cos 13\pi/6) \).
0
Evaluate \( \cos[\cos^{-1}(-\sqrt{3}/2) + \pi/6] \).
-1
Prove that \( \cot(\pi/4 - 2\cot^{-1}3) = 7 \).
7
Find the value of \( \tan^{-1}(-1/\sqrt{3}) + \cot^{-1}(1/\sqrt{3}) + \tan^{-1}(\sin(-\pi/2)) \).
-\pi/12
Find the value of \( \tan^{-1}(\tan 2\pi/3) \).
-\pi/3
Show that \( 2\tan^{-1}(-3) = -\pi/2 + \tan^{-1}(-4/3) \).
-\pi/2 + \tan^{-1}(-4/3)
Find the real solutions of the equation \( \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^{2}+x+1} = \pi/2 \).
0, -1
Find the value of the expression \( \sin(2\tan^{-1}(1/3)) + \cos(\tan^{-1}(2\sqrt{2})) \).
14/15
If \( 2\tan^{-1}(\cos \theta) = \tan^{-1}(2\csc \theta) \), then show that \( \theta = \pi/4 \).
\pi/4
Show that \( \cos(2\tan^{-1}(1/7)) = \sin(4\tan^{-1}(1/3)) \).
Both sides are equal.
Solve the equation \( \cos(\tan^{-1} x) = \sin(\cot^{-1}(3/4)) \).
-3/4, 3/4
Prove that \(\tan^{-1}\left(\dfrac{\sqrt{1+x^{2}} + \sqrt{1-x^{2}}}{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}\right) = \dfrac{\pi}{4} + \dfrac{1}{2} \cos^{-1} x^{2}\).
Find the simplified form of \(\cos^{-1}\left(\dfrac{3}{5}\cos x + \dfrac{4}{5}\sin x\right)\), where \(x \in \left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]\).
\(\tan^{-1}\dfrac{4}{3} - x\)
Prove that \(\sin^{-1}\dfrac{8}{17} + \sin^{-1}\dfrac{3}{5} = \sin^{-1}\dfrac{77}{85}\).
\(\dfrac{77}{85}\)
Show that \(\sin^{-1}\dfrac{5}{13} + \cos^{-1}\dfrac{3}{5} = \tan^{-1}\dfrac{63}{16}\).
\(\dfrac{63}{16}\)
Prove that \(\tan^{-1}\dfrac{1}{4} + \tan^{-1}\dfrac{2}{9} = \sin^{-1}\dfrac{1}{\sqrt{5}}\).
\(\dfrac{1}{\sqrt{5}}\)
Find the value of \(4 \tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{239}\).
\(\dfrac{\pi}{4}\)
Show that \(\tan\left(\dfrac{1}{2}\sin^{-1}\dfrac{3}{4}\right) = \dfrac{4 - \sqrt{7}}{3}\) and justify why the other value \(\dfrac{4 + \sqrt{7}}{3}\) is ignored.
\(\dfrac{4 - \sqrt{7}}{3}\)
If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) is an arithmetic progression with common difference \(d\), evaluate the expression:
\[ \tan \left[ \tan^{-1}\left(\dfrac{d}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{2}a_{3}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{3}a_{4}}\right) + \cdots + \tan^{-1}\left(\dfrac{d}{1 + a_{n-1}a_{n}}\right) \right] \]
\(\dfrac{a_{n} - a_{1}}{1 + a_{1}a_{n}}\)
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
The principal value of \(\cos^{-1}(-\dfrac{1}{2})\) is ____.
\(\dfrac{2\pi}{3}\)
The value of \(\sin^{-1}\left(\sin \dfrac{3\pi}{5}\right)\) is ____.
\(\dfrac{2\pi}{5}\)
If \(\cos(\tan^{-1} x + \cot^{-1} \sqrt{3}) = 0\), then value of \(x\) is ____.
\(\sqrt{3}\)
The set of values of \(\sec^{-1}\left(\dfrac{1}{2}\right)\) is ____.
\(\varphi\)
The principal value of \(\tan^{-1} \sqrt{3}\) is ____.
\(\dfrac{\pi}{3}\)
The value of \(\cos^{-1}(\cos \dfrac{14\pi}{3})\) is ____.
\(\dfrac{2\pi}{3}\)
The value of \(\cos(\sin^{-1} x + \cos^{-1} x)\), \(|x| \le 1\), is ____.
0
The value of the expression \(\tan\left(\dfrac{\sin^{-1} x + \cos^{-1} x}{2}\right)\), when \(x = \dfrac{\sqrt{3}}{2}\), is ____.
1
If \(y = 2 \tan^{-1} x + \sin^{-1}\left(\dfrac{2x}{1 + x^{2}}\right)\) for all \(x\), then ____ < y < ____.
-2\pi
2\pi
The result \(\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\dfrac{x - y}{1 + xy}\right)\) is true when value of \(xy\) is ____.
\(xy > -1\)
The value of \(\cot^{-1}(-x)\) for all \(x \in \mathbb{R}\) in terms of \(\cot^{-1} x\) is ____.
\(\pi - \cot^{-1} x\)
All trigonometric functions have inverse over their respective domains.
False
The value of the expression \((\cos^{-1} x)^2\) is equal to \(\sec^{2} x\).
False
The domain of trigonometric functions can be restricted to any one of their branches (not necessarily principal value) in order to obtain their inverse functions.
True
The least numerical value, either positive or negative, of angle \(\theta\) is called the principal value of the inverse trigonometric function.
True
The graph of an inverse trigonometric function can be obtained from the graph of its corresponding trigonometric function by interchanging \(x\) and \(y\) axes.
True
The minimum value of \(n\) for which \(\tan^{-1} \dfrac{n}{\pi} = \dfrac{\pi}{4}\), \(n \in \mathbb{N}\), is valid is 5.
False
The principal value of \(\sin^{-1}\left[\cos\left(\sin^{-1} \dfrac{1}{2}\right)\right]\) is \(\dfrac{\pi}{3}\).
True