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  4. A quadratic equation whose roots are sec2 α and cosec2 α can be

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Q. A quadratic equation whose roots are $\sec^2 \alpha$ and $cosec^2$ $ \alpha$ can be

Trigonometric Functions

Solution:

If $\sec^2 \, \alpha$ and $cosec^2 \, \alpha$ are the roots of $x^2 + ax + b = 0$, then $\sec^2 \, \alpha + cosec^2 \, \alpha = - a$ and $\sec^2 \, \alpha \, cosec^2 \, \alpha = b$
$\Rightarrow \, -a = b$
$(\because \, \sec^2 \, \alpha \, cosec^2 \, \alpha = \sec^2 \, \alpha + cosec^2 \, \alpha)$
$\Rightarrow \, a + b = 0$
Also $\sec^2 \, \alpha + cosec^2 \alpha \ge 4$
$\Rightarrow \, b = \sec^2 \, \alpha \, cosec^2 \, \alpha \ge 4$
Hence, $x^2 - 5x + 5 = 0$ can be a quadratic whose roots are $cosec^2 \, \alpha $ and $sec^2 \alpha$.