Considering only the principal values of inverse trigonometric functions,the number of positive real values of $x$ satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is:

If $\tan^{-1} (x+ 2)+ \tan^{- 1}( x -2)= \tan^{-1} (\frac{1}{2}),$ then the sum of the value$(s)$ of $x$ is equal to-

Consider the following statements:
Assertion $(A)$: For $x \in \mathbb{R}-\{1\}$, $\frac{d}{dx}\left(\tan^{-1}\left(\frac{1+x}{1-x}\right)\right) = \frac{d}{dx}\left(\tan^{-1} x\right)$.
Reason $(R)$: For $x < 1$, $\tan^{-1}\left(\frac{1+x}{1-x}\right) = \frac{\pi}{4} + \tan^{-1} x$, and for $x > 1$, $\tan^{-1}\left(\frac{1+x}{1-x}\right) = -\frac{3\pi}{4} + \tan^{-1} x$.
The correct answer is:

If $\tan ^{-1}\left(\frac{1-x}{1+x}\right)-\frac{1}{2} \tan ^{-1} x=0$,for $x>0$,then $x=$

Let $f(\theta) = \sin ( \tan ^{-1} ( \frac{\sin \theta}{\sqrt{\cos 2 \theta}} ) )$,where $-\frac{\pi}{4} < \theta < \frac{\pi}{4}$,then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is

With reference to the principal values,if $\sin ^{-1} x + \sin ^{-1} y + \sin ^{-1} z = \frac{3 \pi}{2}$,then $x^{100} + y^{100} + z^{100} =$