Inverse Trigonometric Functions

For \(k \in \mathbb{R}\), let the solutions of the equation \(\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0<|x|<\frac{1}{\sqrt{2}}\) be \(\alpha\) and \(\beta\), wher...

The domain of the function \(f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)\), where [t] is the greatest integer function, is :

If the inverse trigonometric functions take principal values then $${\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right...

\(\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)\) is equal to :

If \(0 < x < {1 \over {\sqrt 2 }}\) and \({{{{\sin }^{ - 1}}x} \over \alpha } = {{{{\cos }^{ - 1}}x} \over \beta }\), then the value of \(\sin \left( {{{2\pi \alpha } \over {\alpha + \beta }}} \right)\) is :

The value of \({\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)\) is equal to :

Let \(x = \sin (2{\tan ^{ - 1}}\alpha )\) and \(y = \sin \left( {{1 \over 2}{{\tan }^{ - 1}}{4 \over 3}} \right)\). If \(S = \{ a \in R:{y^2} = 1 - x\}\), then \(\sum\limits_{\alpha \in S}^{} {16{\alpha ^3}}\) is equal to _______________...

The domain of the function \(f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}\) is :

The set of all values of k for which \({({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R\), is the interval :

Let \(x * y = {x^2} + {y^3}\) and \((x * 1) * 1 = x * (1 * 1)\).
Then a value of \(2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)\) is :

The domain of the function\(f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)\) is :

If \({({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a\); 0 < x < 1, a \(\ne\) 0, then the value of 2x2 \(-\) 1 is :

Let M and m respectively be the maximum and minimum values of the function f(x) = tan\(-\)1 (sin x + cos x) in \(\left[ {0,{\pi \over 2}} \right]\), then the value of tan(M \(-\) m) is equal to :

If \({{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}\); \(0 < x < 1\), then the value of \(\cos \left( {{{\pi c} \over {a + b}}} \right)\) is :

If 0 < a, b < 1, and tan\(-\)1a + tan\(-\)1b = \({\pi \over 4}\), then the value of \((a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....\) ...

If \(\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p}\), then the value of tan p is :

The domain of the function \({{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)\) is :

cosec\(\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]\) is equal to :

A possible value of \(\tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right)\) is :

If the domain of the function \(f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}\) is the interval (\(\alpha\), \(\beta\)], then \(\alpha\) + \(\beta\) is equal to :

The number of real roots of the equation \({\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}\) is :

The value of \(\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right)\) is equal to :

\({\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))\) is equal to :(The inverse trigonometric functions take the principal values)

If cot\(-\)1(\(\alpha\)) = cot\(-\)1 2 + cot\(-\)1 8 + cot\(-\)1 18 + cot\(-\)1 32 + ...... upto 100 terms, then \(\alpha\) is :

The sum of possible values of x for tan\(-\)1(x + 1) + cot\(-\)1\(\left( {{1 \over {x - 1}}} \right)\) = tan\(-\)1\(\left( {{8 \over {31}}} \right)\) is :

The number of solutions of the equation \({\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}\), for x\(\in\)[\(-\)1, 1], and [x] denotes the greatest integer less than or equal to...

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy $${\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x$...

If S is the sum of the first 10 terms of the series
$${\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan ^{ - 1}}\left( {{1 \over {13}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {21}}} \right) +...

2\(\pi\) - \(\left( {{{\sin }^{ - 1}}{4 \over 5} + {{\sin }^{ - 1}}{5 \over {13}} + {{\sin }^{ - 1}}{{16} \over {65}}} \right)\) is equal to :

The domain of the function
f(x) = \({\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)\) is (–
\(\infty\), -a]\(\cup\)[a, \(\infty\)). Then a is equal to :

If \({\cos ^{ - 1}}\left( {{2 \over {3x}}} \right) + {\cos ^{ - 1}}\left( {{3 \over {4x}}} \right) = {\pi \over 2}\) (x > \(3 \over 4\)), then x is equal to :

If  x = sin\(-\)1(sin10) and y = cos\(-\)1(cos10), then y \(-\) x is equal to :

If \(\alpha = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)\), \(\beta = {\tan ^{ - 1}}\left( {{1 \over 3}} \right)\) where \(0 < \alpha ,\beta < {\pi \over 2}\) , then \(\alpha\) - \(\beta\) is equal to :

Considering only the principal values of inverse functions, the set
A = { x \(\ge\) 0: tan\(-\)1(2x) + tan\(-\)1(3x) = \({\pi \over 4}\)}

The value of \({\sin ^{ - 1}}\left( {{{12} \over {13}}} \right) - {\sin ^{ - 1}}\left( {{3 \over 5}} \right)\) is equal to :

All x satisfying the inequality (cot–1
x)2– 7(cot–1 x) + 10 > 0, lie in the interval :

The value of \(\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} \right)\) is :

If \({\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha\),where –1 \(\le\) x \(\le\) 1, – 2 \(\le\) y \(\le\) 2, x \(\le\) \({y \over 2}\)
, then for all x, y, 4x2
– 4xy cos \(\alpha\) + y2
is equal
to :

A value of x satisfying the equation sin[cot−1 (1+ x)] = cos [tan−1 x], is :

The value of tan-1 \(\left[ {{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} } \over {\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right],\) \(\left| x \right| < {1 \over 2},x \ne 0,\) is equal to :

Let \({\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),\)
where \(\left| x \right| < {1 \over {\sqrt 3 }}.\) Then a value of \(y\) is :

If \(x, y, z\) are in A.P. and \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in A.P., then :

The value of \(cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)\) is :

If sin-1\(\left( {{x \over 5}} \right)\) + cosec-1\(\left( {{5 \over 4}} \right)\) = \({\pi \over 2}\), then the value of x is :

If \({\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,\) then \(4{x^2} - 4xy\cos \alpha + {y^2}\) is equal to :

The trigonometric equation \({\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a\) has a solution for :

\({\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,\) then sin x is equal to :