Question

\[ \text{If } f(x)=\cos^2x+\sec^2x, \]

\[ \text{then} \]

(a) \(f(x)<1\)
(b) \(f(x)=1\)
(c) \(1<f(x)<2\)
(d) \(f(x)\ge2\)

Solution

Let

\[ a=\cos^2x \]

Then

\[ \sec^2x=\frac1a \]

So,

\[ f(x)=a+\frac1a \]

Using AM ≥ GM,

\[ a+\frac1a\ge2 \]

Therefore,

\[ f(x)\ge2 \]

Answer

\[ \boxed{f(x)\ge2} \]

Correct Option: (d)