Find the product $\sec x\cdot \csc x$

Question

To be $\frac \pi2 < x < \pi $. If $$\tan^2 x+\cot^2 x +4\tan x + 4\cot x - 10=0,$$ find the product $\sec x\cdot \csc x$.

Answer 1

Hint: $$\tan t+\cot t=\sec t\csc t$$ $$\tan^2t+\cot ^2t=\sec^2 t\csc^2 t-2$$