Write the least value of ^2x+ ^2x. - Vidyadip

Question

Write the least value of $\cos^2\text{x}+\sec^2\text{x}.$

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Answer

$-1\le\cos\text{x}\le1$
$\Rightarrow1\le\cos^2\text{x}\le1$
$-1\le\frac{1}{\cos\text{x}}\le1$
$\Rightarrow1\le\frac{1}{\cos^2\text{x}}\le1$
$\Rightarrow1\le\sec^2\text{x}\le1$
$2\le\cos^2\text{x}+\sec^2\text{x}$.
So the least value of $\cos^2\text{x}+\sec^2\text{x}$ is 2.

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