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Question
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Options
f(x) < 1
f(x) = 1
1 < f(x) < 2
f(x) ≥ 2
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Solution
\[f\left( x \right) = \cos^2 x + \sec^2 x\]
\[ = \cos^2 x + \sec^2 x - 2\cos x\sec x + 2\cos x\sec x\]
\[ = \left( \sec x - \cos x \right)^2 + 2\]
\[ \therefore f\left( x \right) \geq 2 \forall x \left[ \left( \sec x - \cos x \right)^2 \geq 0 \forall x \right]\]
Hence, the correct option is answer f(x) ≥ 2.
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