Is it possible to solve the equation
$$ a \sin x + b \cos x + c \cos^3 x = d \cos x $$
where $c\neq0$ using some coefficients $a$, $b$, $c$, and $d$? I can't see how to make the frequency of oscillation for the left hand side of the equation equal to the frequency for the right hand side.
(Pursuant to Mr. Millikan's observation that $b$ and $d$ can be combined, I now replace the both of them with $e$.) So, $a\sin{x}+e\cos{x}+c\cos^3{x}=0$.
How about moving $a\sin{x}$ to the RHS, then squaring, then substituting $a^2-a^2\cos^2{x}$ for $a^2\sin^2{x}$? This would give you a cubic equation in $\cos^2{x}$. No guarantee, though, that $\cos^2{x}$ would turn out to be a proper value for a squared cosine -- i.e., $\in [0,1]$. It all depends upon the coëfficients.
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$$c^2\cos^6{x}+2ce\cos^4{x}+(e^2+a^2)\cos^2{x}-a^2=0$$