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  4. The value of sec 4 x+ operatornamecosec4 x+ sec 4 x operatornamecosec4 x can be

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Q. The value of $\sec ^4 x+\operatorname{cosec}^4 x+\sec ^4 x \operatorname{cosec}^4 x$ can be

Sequences and Series

Solution:

$ \frac{\sin ^2 x+\cos ^2 x}{2} \geq \sqrt{\sin ^2 x \cos ^2 x} $
$\Rightarrow \sec ^2 x \operatorname{cosec}^2 x \geq 4 \Rightarrow \sec ^4 x \operatorname{cosec}^4 x \geq 16 $....(i)
$\text { Again, } \frac{\sec ^4 x+\operatorname{cosec}^4 x}{2} \geq \sqrt{\sec ^4 x \operatorname{cosec}^4 x}$
$\sec ^4 x+\operatorname{cosec}^4 x \geq 8$....(ii)
$\text { From (i) and (ii), } $
$\sec ^4 x+\operatorname{cosec}^4 x+\sec ^4 x \cdot \operatorname{cosec}^4 x \geq 24$
$\text { (Equality holds when } \sec x=\operatorname{cosec} x= \pm \sqrt{2}$