The number of solutions of

$\sin \left\{{\sin }^{-1}\left({\log }_{1/2}x\right)\right\}+2\left|\cos \left\{{\sin }^{-1}\left(\frac{x}{2}-\frac{3}{2}\right)\right\}\right|=0$, is

The two terms on the LHS of the given equation

are meaningful, if

$\begin{array}{l}-1\le {\log }_{1/2}x\le 1\text{ and }-1\le \frac{x}{2}-\frac{3}{2}\le 1\\ \Rightarrow 2\ge x\ge \frac{1}{2}\text{ and }1\le x\le 5\Rightarrow 1\le x\le 2\end{array}$

Now,

$\begin{array}{l}1\le x\le 2\\ \Rightarrow \frac{1}{2}\le \frac{x}{2}\le 1\\ \Rightarrow -\frac{1}{2}\le \frac{x}{2}-1\le 0\\ \Rightarrow -\frac{\pi }{6}\le {\sin }^{-1}\left(\frac{x}{2}-1\right)\le 0\\ \Rightarrow \frac{\sqrt{3}}{2}\le \cos \left\{{\sin }^{-1}\left(\frac{x}{2}-1\right)\right\}\le 1\\ \Rightarrow \sqrt{3}\le 2\cos \left\{{\sin }^{-1}\left(\frac{x}{2}-1\right)\right\}\le 2\\ \Rightarrow \sqrt{3}\le 2\left|\cos \left\{{\sin }^{-1}\left(\frac{x}{2}-1\right)\right\}\right|\le 2\end{array}$

Again,

$\begin{array}{l}1\le x\le 2\\ \Rightarrow -1\le {\log }_{1/2}x\le 0\\ \Rightarrow -\frac{\pi }{2}\le {\sin }^{-1}\left({\log }_{1/2}x\right)\le 0\\ \Rightarrow -1\le \sin \left\{{\sin }^{-1}\left({\log }_{1/2}x\right)\right\}\le 0\\ \Rightarrow \sqrt{3}-1\le \sin \left\{{\sin }^{-1}\left({\log }_{1/2}x\right\}+\left|\cos {\sin }^{-1}\left(\frac{x}{2}-\frac{3}{2}\right)\right|\le 2\right.\\ \Rightarrow \sin \left\{{\sin }^{-1}\left({\log }_{1/2}x\right)\right\}+2\left|\cos \left\{{\sin }^{-1}\left(\frac{x}{2}-\frac{3}{2}\right)\right\}\right|\end{array}$

$\ne 0$ for any $x$

Hence, the given equation has no solution.