For $\alpha, \beta, \gamma \neq 0$. If $\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$, then $\gamma$ equal to

If the domain of the function f(x)=cos^(-1)⁡((2-|x|)/4)+(log_e⁡(3-x))^(-1) is [-α,β)-{y}, then α+β+γ is equal to :

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec ^{-1}(2[x]+1)$ is :

Considering the principal values of the inverse trigonometric functions, $...

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^{2}\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^{2}\right)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^{2}\right)}{(\gamma-\alpha)}\right\}$ is equal to :

Let $\mathrm{S}=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}$. Then $\sum_{x \in \mathrm{~S}}(2 x-1)^2$ is equal to ______.

$\cos \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\sin }^{ - 1}}\frac{5}{{13}} + {{\sin }^{ - 1}}\frac{{33}}{{65}}} \right)$ is equal to:

If for some $\alpha ,\beta ;\alpha \le \beta ,\alpha + \beta = 8$ and $...

If $\frac{\pi }{2}\le x\le \frac{{3\pi }}{4}$ , then ${{\cos }^{{-1}}}\left( {\frac{{12}}{{13}}\cos x+\frac{5}{{13}}\sin x} \right)$ is equal to