Question

The value of ${\sec }^{4}x+{\mathrm{cosec}}^{4}x+{\sec }^{4}x{\mathrm{cosec}}^{4}x$ can be

• A. 16
• B. 20
• C. 32
• D. 64

Answer 1

Answer: (C), (D)

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