Inverse Trigonometric Functions

The true set of values of ' $K$ ' for which $\sin ^{-1}\left(\frac{1}{1+\sin ^2 x}\right)=\frac{K \pi}{6}$ may have a solution is

If $f(x)=x\left(1331 x^2-3630 x+3300\right)$, then for $a=\cos ^2\left(\tan ^{-1}\left(\sin \left(\cot ^{-1} 3\right)\right)\right)$

If for two real numbers $\mathrm{a}, \mathrm{b}$ with $|\mathrm{a}| \leq 1$ and $|\mathrm{b}| \leq 1$,
$\frac{1}{3}+\frac{\sin ^{-1} a+\sin ^{-1} b}{4}+\frac{\left(\sin ^{-1} a+\sin ^{-1} b\right)^2}{16}+\frac{\left(\sin ^{-1} a+\sin ^{-1} ...

Let $g(x)=a x+b$, where $a<0$ and $g$ is defined from $[1,3]$ onto $[0,2]$. Then the value of $\cot \left(\cos ^{-1}(|\sin x|+|\cos x|)+\right. \left.\sin ^{-1}(-|\cos x|-|\sin x|)\right)$ is equal to

If $\sum\limits_{r=1}^{\infty} \tan ^{-1}\left(\frac{1}{2 r^2}\right)=a$, then $\tan a$ is equal to

If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ is equal to

The number of solutions of $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$ is

For \(y = {\sin ^{ - 1}}\left\{ {{{5x + 12\sqrt {1 - {x^2}} } \over {13}}} \right\};\left| x \right| \le 1\), if \(a(1 - {x^2}){y_2} + bx{y_1} = 0\) then (a, b) =

If \(0 \le A \le {\pi \over 4}\), then \({\tan ^{ - 1}}\left( {{1 \over 2}\tan 2A} \right) + {\tan ^{ - 1}}(\cot A) + {\tan ^{ - 1}}({\cot ^3}A)\)

The possible values of x, which satisfy the trigonometric equation \({\tan ^{ - 1}}\left( {{{x - 1} \over {x - 2}}} \right) + {\tan ^{ - 1}}\left( {{{x + 1} \over {x + 2}}} \right) = {\pi \over 4}\) are

If \(f(x) = {\tan ^{ - 1}}\left[ {{{\log \left( {{e \over {{x^2}}}} \right)} \over {\log (e{x^2})}}} \right] + {\tan ^{ - 1}}\left[ {{{3 + 2\log x} \over {1 - 6\log x}}} \right]\), then the value of f''(x) is equal to

If \({r^2} = {x^2} + {y^2} + {z^2}\), then prove that \({\tan ^{ - 1}}\left( {{{yz} \over {rx}}} \right) + {\tan ^{ - 1}}\left( {{{zx} \over {ry}}} \right) + {\tan ^{ - 1}}\left( {{{xy} \over {rz}}} \right) = {\pi \over 2}\)

The solution set of the inequation \({\cos ^{ - 1}}x < {\sin ^{ - 1}}x\) is

Value of \({\tan ^{ - 1}}\left( {{{\sin 2 - 1} \over {\cos 2}}} \right)\) is

\(\tan \left[ {{\pi \over 4} + {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right] + \tan \left[ {{\pi \over 4} - {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right]\) is equal to

The value of \(\tan \alpha + 2\tan (2\alpha ) + 4\tan (4\alpha ) + ... + {2^{n - 1}}\tan ({2^{n - 1}}\alpha ) + {2^n}\cot ({2^n}\alpha )\) is