The value of $\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right\}$ is

Domain of definition of the real valued function $f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}}$ is

$$\frac{\mathrm{d}}{\mathrm{~d} x}\left(\cos ^{-1}\left(\frac{x-\frac{1}{x}}{x+\frac{1}{x}}\right)\right)=$$

The approximate value of $\tan ^{-1}(0.999)$ is (use $\pi=3.1415$ )