The number of real solutions of the equation $\sin^{-1}\left(\sum_{i=1}^{\infty} x^{i+1} - x\sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}\right) = \frac{\pi}{2} - \cos^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i} - \sum_{i=1}^{\infty}(-x)^{i}\right)$ lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is ______. (Here, the inverse trigonometric functions $\sin^{-1}x$ and $\cos^{-1}x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

$S_1 = \sum_{i=1}^{\infty} x^{i+1} - x\sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}$ $S_2 = \sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i} - \sum_{i=1}^{\infty}(-x)^{i}$ $\sin^{-1}(S_1) = \frac{\pi}{2} - \cos^{-1}(S_2),\quad x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$ $\text{To find: number of real solutions in } \left(-\frac{1}{2}, \frac{1}{2}\right)$ $\text{Plan: }1.\ \text{evaluate } S_1, S_2$ $2.\ \sin^{-1}t + \cos^{-1}t = \frac{\pi}{2} \Rightarrow S_1 = S_2$ $3.\ \text{count roots in } \left(-\frac{1}{2}, \frac{1}{2}\right)$ $\sum_{i=1}^{\infty} x^{i+1} = \frac{x^{2}}{1-x},\qquad x\sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i} = x\cdot\frac{x/2}{1-x/2} = \frac{x^{2}}{2-x}$ $S_1 = \frac{x^{2}}{1-x} - \frac{x^{2}}{2-x} = \frac{x^{2}}{(1-x)(2-x)}$ $\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i} = \frac{-x/2}{1+x/2} = \frac{-x}{2+x},\qquad \sum_{i=1}^{\infty}(-x)^{i} = \frac{-x}{1+x}$ $S_2 = \frac{-x}{2+x} + \frac{x}{1+x} = \frac{x}{(1+x)(2+x)}$ $\sin^{-1}t + \cos^{-1}t = \frac{\pi}{2} \Rightarrow \frac{\pi}{2} - \cos^{-1}(S_2) = \sin^{-1}(S_2)$ $\sin^{-1}(S_1) = \sin^{-1}(S_2) \Rightarrow S_1 = S_2$ $x^{2}(1+x)(2+x) = x(1-x)(2-x)$ $x = 0 \quad \text{or} \quad x(1+x)(2+x) = (1-x)(2-x)$ $\Rightarrow x^{3} + 2x^{2} + 5x - 2 = 0$ $f(x) = x^{3}+2x^{2}+5x-2,\qquad f'(x) = 3x^{2}+4x+5 > 0\ \ \forall x$ $f(0) = -2 < 0,\qquad f\!\left(\tfrac{1}{2}\right) = \tfrac{9}{8} > 0$ $\Rightarrow \text{exactly one root in } \left(0, \tfrac{1}{2}\right)$ $x = 0 \quad \text{and} \quad x \approx 0.344$ $\boxed{\text{Number of real solutions} = 2}$