Q:

Number of solutions of $\tan^2 x - \sec^{10} x +1=0$ in the interval $(0,10)$

How many solutions of the equation $$\tan^2 x - \sec^{10} x +1=0$$ lie in the interval $(0,10)$?

A:

Hint:  Since $\tan^2x + 1 = \sec^2x$, 
\begin{align*}
\tan^2x - \sec^{10}x + 1 & = 0\\
\sec^2x - \sec^{10}x & = 0\\
\sec^2x(1 - \sec^8x) & = 0
\end{align*}
Since $\sec^2x \neq 0$, you need to find the solutions of $1 - \sec^8x = 0$ in the interval $(0, 10)$.  

