Solution:

Option (D) $f(x) \geq 2$ is correcct.

Explanation:

As per the question,

$f(x)=\cos ^{2} x+\sec ^{2} x$

It is known that, A.M $\geq$ G.M.

$\begin{array}{l}
\Rightarrow \frac{\cos ^{2} x+\sec ^{2} x}{2} \geq \sqrt{\cos ^{2} x \sec ^{2} x} \\
\Rightarrow \frac{\cos ^{2} x+\sec ^{2} x}{2} \geq \sqrt{\cos ^{2} x \frac{1}{\cos ^{2} x}} \\
\Rightarrow \frac{\cos ^{2} x+\sec ^{2} x}{2} \geq 1 \\
\Rightarrow \cos 0^{2} x+\sec ^{2} x \geq 2 \\
\Rightarrow f(x) \geq 2
\end{array}$

As a result, option (D) $f(x) \geq 2$ is the correct answer.