Does this propertly holds for inverse too? (part-lll)

arcCosec(x) is the same as arcSin(1/x). therefore arcSin(x) * arcSin(1/x)=1 for which the is no possible solution because when x<1, 1/x >1 and vice versa and having know that Sinx less than or equal to one

It would help to add in the question that $x$ must be real. But moving on to the solution: The equation is equivalent to $\frac{1}{\arcsin{x}}=\arcsin{\frac{1}{x}}$ . Since the argument of arc sine must be less than or equal to $1$ in absolute value, we must have $|x|\leq 1$ and $\frac{1}{|x|}\leq 1$ . These inequalities are only simultaneously satisfied when $x=-1,1$ . But testing these in the equation, we see that neither are solutions. Therefore, there are no solutions.