Question

How can I find all the solutions in the interval (0,2pi) with the equation sec^2x+5tanx=-2?

Answer 1

use the identity\[\sec^2x-1=\tan^2x\]so we convert\[\sec^2x+5\tan x=-2\]\[\sec^2x-1+5\tan x=-3\]\[\tan^2x+5\tan x=-3\]\[\tan^2x+5\tan x+3=0\]which is quadratic in tanx, though it doesn't factor (bummer)
you can take it from here hopefully