Question

Find the number of solution of the equation ${\tan }^{2}x-{\sec }^{10}x+1=0$ in $(0,10)$

Answer 1

Given, ${\tan }^{2}x-{\sec }^{10}x+1=0\Rightarrow {\sec }^{2}x-{\sec }^{10}x=0\Rightarrow {\sec }^{2}x(1-{\sec }^{8}x)=0$
$\Rightarrow {\sec }^{2}x=0$ or $1-{\sec }^{8}x=0$
$\Rightarrow$ not possible or ${\sec }^{8}x=1$
$\Rightarrow \sec x=\pm 1$
$\Rightarrow x=2n\pi \pm \frac{\pi }{2},2n\pi \pm \frac{3\pi }{2}$