Inverse Trigonometric Functions

Considering only the principal values of the inverse trigonometric functions, the value of \(\cot^{-1}(\cot(-11)) + 10 \sin\left(2 \cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin(2 \tan^{-1}(2))\)is

The total number of real solutions of the equation
$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1}\left(\frac{6 \tan \theta}{9 + \tan^2 \theta}\right) $
is
(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\tan^{-1} ...

Considering only the principal values of the inverse trigonometric functions, the value of

\(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\)

is

For any $y \in \mathbb{R}$, let $\cot ^{-1}(y) \in(0, \pi)$ and $\tan ^{-1}(y) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the sum of all the solutions of the equation $\tan ^{-1}\left(\frac{6 y}{9-y^2}\right)+\cot ^{-1}\left(\frac...

Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for $x \in \mathbb{R}$. Then the number of real solutions of the equation $\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1}(\tan x)$ in the set $\left(-\frac{3 \pi}{2},-\frac{\pi}{2}...

Considering only the principal values of the inverse trigonometric functions, the value of

\(\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^{2}}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^{2}}+\tan ^{-1} \frac{\sqrt{2}}{\pi}\)

is

The value of $${\sec ^{ - 1}}\left( \matrix{
{1 \over 4}\sum\limits_{k = 0}^{10} {\sec \left( {{{7\pi } \over {12}} + {{k\pi } \over 2}} \right)}
\sec \left( {{{7\pi } \over {12}} + {{(k + 1)\pi } \over 2}} \right) \hfill \cr} \right)...

For any positive integer n, define \({f_n}:(0,\infty ) \to R\) as\({f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)\)for all x\(\in\)(0, \(\infty\)). (Here, the inverse trigonometric fu...

The number of real solutions of the equation $$\eqalign{
& {\sin ^{ - 1}}\left( {\sum\limits_{i = 1}^\infty {} {x^{i + 1}} - x\sum\limits_{i = 1}^\infty {} {{\left( {{x \over 2}} \right)}^i}} \right) \cr
& = {\pi \over 2} - {\cos ...

If \(\alpha\) \(= 3{\sin ^{ - 1}}\left( {{6 \over {11}}} \right)\) and \(\beta = 3{\cos ^{ - 1}}\left( {{4 \over 9}} \right),\) where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are...

Match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) List-\(I\)
(P.)\(\,\,\,\,\) Let $$y\left( x \right) = \cos \left( {3{{\cos }^{ - 1}}x} \right),x \i...

Let f : [0, 4\(\pi\)] \(\to\) [0, \(\pi\)] be defined by f(x) = cos\(-\)1 (cos x). The number of points x \(\in\) [0, 4\(\pi\)] satisfying the equation \(f(x) = {{10 - x} \over {10}}\) is

Match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
List \(I\)
\(P.\)\(\,\,\,\,\,\) $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }...

The value of \(\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} \right)\) is

If \(0 < x < 1\), then
\(\sqrt {1 + {x^2}} {\left[ {{{\left\{ {x\cos \left( {{{\cot }^{ - 1}}x} \right) + \sin \left( {{{\cot }^{ - 1}}x} \right)} \right\}}^2} - 1} \right]^{1/2}} =\)

Let \((x,y)\) be such that \({\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}\).
Match the statements in Column I with the statements in Column II.

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{borde...

Let F(x) be an indefinite integral of \(\sin^2x\).
Statement 1 : The function F(x) satisfies F(\(x+\pi\)) = F(\(x\)) for all real x.
Statement 2 : \({\sin ^2}(x + \pi ) = {\sin ^2}x\) for all real x.

Let \((x, y)\) be such that \({\sin ^{ - 1}}\left( {ax} \right) + {\cos ^{ - 1}}\left( y \right) + {\cos ^{ - 1}}\left( {bxy} \right) = {\pi \over 2}\).
Column \(I\)
(A) If \(a=1\) and \(b=0,\) then \((x, y)\)
(B) If \(a=1\) and \(b=1,\) t...

The value of \(x\) for which \(sin\left( {{{\cot }^{ - 1}}\left( {1 + x} \right)} \right) = \cos \left( {{{\tan }^{ - 1}}\,x} \right)\) is

Prove that \(\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}}\).

If \({\sin ^{ - 1}}\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 4} - ....} \right)\)
\($ + {\cos ^{ - 1}}\left( {{x^2} - {{{x^4}} \over 2} + {{{x^6}} \over 4} - ....} \right) = {\pi \over 2}\)$
for \(0 < \left| x \right| < \sqrt 2 ,\) ...

The number of real solutions of
\({\tan ^{ - 1}}\,\,\sqrt {x\left( {x + 1} \right)} + {\sin ^{ - 1}}\,\,\sqrt {{x^2} + x + 1} = \pi /2\) is

If we consider only the principle values of the inverse trigonometric functions then the value of
\(\tan \left( {{{\cos }^{ - 1}}{1 \over {5\sqrt 2 }} - {{\sin }^{ - 1}}{4 \over {\sqrt {17} }}} \right)\) is

The greater of the two angles \(A = 2{\tan ^{ - 1}}\left( {2\sqrt 2 - 1} \right)\) and \(B = 3{\sin ^{ - 1}}\left( {1/3} \right) + {\sin ^{ - 1}}\left( {3/5} \right)\) is ________ .

The principal value of \({\sin ^{ - 1}}\left( {\sin {{2\pi } \over 3}} \right)\) is

The numerical value of \(\tan \left\{ {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) - {\pi \over 4}} \right\}\) is equal to __________

Find all the solution of \(4\) \({\cos ^2}x\sin x - 2{\sin ^2}x = 3\sin x\)

The value of \(\tan \left[ {{{\cos }^{ - 1}}\left( {{4 \over 5}} \right) + {{\tan }^{ - 1}}\left( {{2 \over 3}} \right)} \right]\) is

Find the value of : \(\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)\) at \(x = {1 \over 5}\), where
\(0 \le {\cos ^{ - 1}}x \le \pi\) and \(- \pi /2 \le {\sin ^{ - 1}}x \le \pi /2\).

Let \(a, b, c\) be positive real numbers Let
\(\theta = {\tan ^{ - 1}}\sqrt {{{a\left( {a + b + c} \right)} \over {bc}}} + {\tan ^{ - 1}}\sqrt {{{b\left( {a + b + c} \right)} \over {ca}}}\) $$ + {\,\,\tan ^{ - 1}}\sqrt {{{c\left( {a + b...