Step-by-step Solution:

\[Step 1:\] Checking the continuity of \( f(x) = \tan^2 x \) at \( x = \frac{\pi}{2} \) - The function \( f(x) = \tan^2 x \) involves the tangent function, which has vertical asymptotes at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots \). - At \( x = \frac{\pi}{2} \), \( \tan x \) is undefined because \( \tan x \to \infty \) as \( x \to \frac{\pi}{2} \). - Since \( \tan^2 x \) is also undefined at \( x = \frac{\pi}{2} \), the function \( f(x) \) is not continuous at \( x = \frac{\pi}{2} \). Thus, Assertion A is false. \[Step 2:\] Checking the continuity of \( g(x) = x^2 \) at \( x = \frac{\pi}{2} \) - The function \( g(x) = x^2 \) is a polynomial function, and polynomials are continuous everywhere. - Since \( g(x) = x^2 \) is defined and continuous at \( x = \frac{\pi}{2} \), we conclude that Reason R is true. --- \[Step 3:\] Evaluating the Relationship Between A and R - \( A \) is false. - \( R \) is true. - \( R \) does not explain \( A \) because \( A \) itself is incorrect. Thus, the correct option is: \[ \boxed{D} \quad \text{(A is false but R is true)} \]