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I'm trying to solve this equation:

$$ f(x).cos^2(ux)+g(x).sin^2(ux) = k; k>0, u>0, x>0 $$ $f$ and $g$ are both affine functions.

I have tried many approaches with trigonometric formulas but I'm stuck.

Can you please give me an hint?

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    By "affine function" you mean $f(x)$ and $g(x)$ are of the form $mx+c$ with $m$ and $c$ constants? – 2017-02-08
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    @ParclyTaxel Yes. More over $m$ and $c$ are strictly positive. – 2017-02-08
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    Unfortunately there is no closed-form way to solve these kinds of equations where a polynomial is multiplied with a trigonometric expression. Sure, we can get the equation down to $(k-f(x))/(g(x)-f(x))=\sin^2(ux)$ but then we cannot manipulate it further - even $\cos(x)=x$ has a transcendental solution. In these kinds of situations we use numerical methods like Newton's to find $x$. – 2017-02-08
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    @ParclyTaxel Thanks for your comment. It IS the response I was looking for. Can you please turn your comment into a response? So we can close this question. – 2017-02-08
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    Converted to an answer. – 2017-02-08

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We could reduce the equation as follows: $$f(x)\cos^2ux+g(x)\sin^2ux=k$$ $$f(x)(1-\sin^2ux)+g(x)\sin^2ux=k$$ $$(g(x)-f(x))\sin^2ux=k-f(x)$$ $$\sin^2ux=\frac{k-f(x)}{g(x)-f(x)}$$ Now, however, we have a trigonometric expression on one side and a rational one on the other. In general there does not exist a closed-form solution to such an equation, so numerical methods are employed in this case.