Question

The number of solutions of the equations $ta{n}^{2}x-se{c}^{10}x+1=0$ in (0, 10) is

• A. 3
• B. 6
• C. 10
• D. 0

Answer 1

Answer: (A)

3

${\sec }^{2}x-{\sec }^{10}x=0$
$\Rightarrow {\sec }^{2}x(1-{\sec }^{8}x)=0$
$\Rightarrow 1-{\sec }^{8}x=0$ as ${\sec }^{2}x\ge 1$
$\Rightarrow {\sec }^{8}x=1$
$\Rightarrow {\cos }^{8}x=1$
$\Rightarrow \cos x=\pm 1$
$\Rightarrow x=n\pi ,n\in I$
$\Rightarrow x=\{\pi ,2\pi ,3\pi \}\in (0,10)$
So the number of solutions is 3.