The number of all values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfying…
The number of all values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfying the equation $(1-\tan \theta)(1+\tan \theta) \sec ^2 \theta+2 \tan ^2 \theta=0$ is
1
0
2
infinitely many.
Solution
$\begin{aligned}
& (1-\tan \theta)(1+\tan \theta) \sec ^2 \theta+2 \tan ^2 \theta=0 \\
& \Rightarrow\left(1-\tan ^2 \theta\right)\left(1+\tan ^2 \theta\right)+2 \tan ^2 \theta=0
\end{aligned}$
Put $\tan ^2 \theta=x$
$\begin{aligned}
& \Rightarrow(1-x)(1+x)+2 x=0 \\
& \Rightarrow 1-x^2+2 x=0 \\
& \Rightarrow x^2-1=2 x
\end{aligned}$
Let us draw graph of $y=x^2-1$ and $y=2 x$
From the graph the two curves are intersecting at 2 points.
$\therefore \quad$ There are 2 values of $x$.
Only one value of $x$ exists, for $\theta \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
$\therefore \quad x=\tan ^2 \theta$
$\therefore \quad$ Two values of $\theta$ satisfies above equation
