Does an elegant solution exist for this trigonometric equation?

Question

I'm trying to solve this:

$\cos ^{-2}x + A\tan{x} = B$

Wolfram alpha spits out an incredibly long and convoluted solution for x.

Is there no simple, straightforward analytical way to solve this?

Answer 1

Note that $\cos^{-2} x=\sec^2 x=1+\tan^2 x$. So we get a quadratic equation in $\tan x$. Solve for $\tan x$ using the Quadratic Formula.

To get $x$, use the $\arctan$ function, remembering that if $x$ is a solution of the equation, then so is $x+n\pi$ for any integer $n$.

Remark: I do not think the solution above qualifies as elegant. The more appropriate term is mechanical. There probably is no elegant solution.

Answer 2

$$1+\tan^2 x+A\tan x=\cos^{-2}x + A\tan{x} = B$$ $$(\tan x) ^2 + A\tan{x} + (1 - B) = 0$$ $$\tan x = \text{two solutions of quadratic equation}$$ $$x=\arctan(\cdots)$$