The range of the expression $f\left(\theta \right)=\frac{{\tan }^2\theta -2\tan \theta -8}{{\tan }^2\theta -4\tan \theta -5}$

Let $y=\frac{{\tan }^2\theta -2\tan \theta -8}{{\tan }^2\theta -4\tan \theta -5}$

$\begin{array}{l}y=\frac{t^2-2t-8}{t^2-4t-5}\\ (y-1)t^2+2(1-2y)t+(8-5y)=0\end{array}$

Clearly, D $\ge$0

$\begin{array}{l}4(1-2y{)}^2-4(y-1\left)\right(8-5y)\ge 0\\ (1-2y{)}^2-(y-1\left)\right(8-5y)\ge 0\\ 9y^2-17y+9\ge 0\end{array}$

Clearly, D < 0 Thus it is true for all y Hence, the range = R