The number of solutions of $sin \begin{Bmatrix} sin^{-1}\left(log_{1/2}x\right)\end{Bmatrix} + 2\begin{vmatrix}cos \begin{Bmatrix}sin^{-1}\left(\frac{x}{2}-\frac{3}{2}\right)\end{Bmatrix}\end{vmatrix}=0$, is

2

The two terms on the LHS of the given equation are meaningful, if

$-1 ≤ log_{1/2}x ≤ 1 $ and $ -1 ≤ \frac{x}{2}-\frac{3}{2}≤1$

$⇒ 2 ≥ x ≥ \frac{1}{2}$ and $ 1 ≤ x ≤ 5 ⇒ 1 ≤ x ≤ 2 $

Now,

$1 ≤ x ≤ 2 $

$⇒ \frac{1}{2}≤ \frac{x}{2} ≤1$

$⇒ -\frac{1}{2}≤ \frac{x}{2}-1 ≤0$

$⇒ -\frac{\pi}{6} ≤ siN^{-1} \left(\frac{x}{2}-1\right) ≤ 0$

$⇒  \frac{\sqrt{3}}{2}≤ cos \begin{Bmatrix}sin^{-1}\left(\frac{x}{2}-1\right)\end{Bmatrix}≤1$

$⇒ \sqrt{3} ≤2 cos \begin{Bmatrix}sin^{-1}\left(\frac{x}{2}-1\right)\end{Bmatrix}≤2$

$⇒ \sqrt{3} ≤2 cos \begin{vmatrix}sin^{-1}\left(\frac{x}{2}-1\right)\end{vmatrix}≤2$

Again,

$1 ≤x ≤2$

$⇒ -1 ≤ log_{1/2}x ≤ 0 $

$⇒ -\frac{\pi}{2} ≤ sin^{-1}(log_{1/2}x) ≤ 0 $

$⇒ -1 ≤ \begin{Bmatrix}sin^{-1}(log_{1/2} x)\end{Bmatrix}≤ 0 $

$⇒ \sqrt{3}-1 ≤sin \begin{Bmatrix}sin^{-1} (log_{1/2}) x\end{Bmatrix} + \begin{vmatrix}cos \, sin^{-1}\left(\frac{x}{2}-\frac{3}{2}\right)\end{vmatrix} ≤ 2$

$⇒ sin \begin{Bmatrix}sin^{-1}(log_{1/2} x)\end{Bmatrix}+ 2\begin{vmatrix} cos \begin{Bmatrix} sin^{-1}\left(\frac{x}{2}-\frac{3}{2}\right) \end{Bmatrix}\end{vmatrix}$ ≠ 0 for any x.

Hence, the given equations has no solution.