The expression given is $\sec^{-1}\left(\frac{\sqrt{3}}{2}\right)$. The domain of the function $f(x) = \sec^{-1}(x)$ is $(-\infty, -1] \cup [1, \infty)$. This means $|x| \geq 1$. Here, $x = \frac{\sqrt{3}}{2} \approx 0.866$. Since $0.866 < 1$, the value $\frac{\sqrt{3}}{2}$ does not lie in the domain of the secant inverse function. Therefore, $\sec^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is undefined, making the set of values a null set. Assertion (A) is True.
The reason states that $\sec^{-1}(x)$ is defined for $x \in \mathbb{R} - (-1, 1)$. This set is equivalent to $(-\infty, -1] \cup [1, \infty)$. This is the correct definition of the domain for the inverse secant function. Reason (R) is True.
Assertion (A) is true because the input $\frac{\sqrt{3}}{2}$ falls outside the domain defined in Reason (R). Thus, Reason (R) is the correct explanation for Assertion (A).
Final Answer: Both A and R are true and R is the correct explanation of A.
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