The number of solutions ofsin⁡sin−1⁡log1/2⁡x+2cos⁡sin−1⁡x2−32=0, is

The two terms on the LHS of the given equationare meaningful, if−1≤log1/2⁡x≤1 and −1≤x2−32≤1⇒ 2≥x≥12 and 1≤x≤5⇒1≤x≤2Now,1≤x≤2⇒12≤x2≤1⇒−12≤x2−1≤0⇒−π6≤sin−1⁡x2−1≤0⇒32≤cos⁡sin−1⁡x2−1≤1⇒3≤2cos⁡sin−1⁡x2−1≤2⇒3≤2cos⁡sin−1⁡x2−1≤2Again,1≤x≤2⇒−1≤log1/2⁡x≤0⇒−π2≤sin−1⁡log1/2⁡x≤0⇒−1≤sin⁡sin−1⁡log1/2⁡x≤0⇒3−1≤sin⁡sin−1⁡log1/2⁡x+cos⁡sin−1⁡x2−32≤2⇒ sin⁡sin−1⁡log1/2⁡x+2cos⁡sin−1⁡x2−32≠0 for any xHence, the given equation has no solution.