If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$,then the value of $x^{2025}+x^{2026}+x^{2027}$ is

The value of $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$ is

Given $0 \leq x \leq \frac{1}{2}$,then the value of $\tan \left[\sin ^{-1}\left\{\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^{2}}}{\sqrt{2}}\right\}-\sin ^{-1} x\right]$ is:

The value of $\cos \left(\tan ^{-1}\left(\sin \left(\cot ^{-1} x\right)\right)\right)$ is

Let $S_{k} = \sum_{r=1}^{k} \tan^{-1}\left(\frac{6^{r}}{2^{2r+1} + 3^{2r+1}}\right)$. Then $\lim_{k \rightarrow \infty} S_{k}$ is equal to

The value of $\sin^{-1}\left(\frac{2\sqrt{2}}{3}\right) + \sin^{-1}\left(\frac{1}{3}\right)$ is equal to