Eliminating $\theta$ from $\cos^3\theta +a\cos\theta =b$ and $\sin^3\theta +a\sin\theta =c$ 
Eliminate $\theta$ from the equations.
$$\cos^3\theta +a\cos\theta =b$$
$$\sin^3\theta +a\sin\theta =c$$

Can anyone solve this question?

So, if I understood properly your question, we have
$$
\left\{ \begin{array}{l}
 \theta ,a,b,c \in R \\ 
 \cos ^3 \theta  + a\cos \theta  - b = 0 \\ 
 \sin ^3 \theta  + a\sin \theta  - c = 0 \\ 
 \end{array} \right.
$$
and we are looking for which domain of the four variables $  \theta ,a,b,c$
the system of the two equations has one or more (real) solutions.
Both equations are depressed cubics.
The discriminants are
$$
\begin{array}{l}
 \Delta _1  =  - 4a^3  - 27b^2  \\  \Delta _2  =  - 4a^3  - 27c^2  \\ 
 \end{array}
$$
If

*

*$\Delta >0$ the equation has three distinct real roots;

*$\Delta = 0$ all roots are still real , and if $a=0$ gives three coincident roots,
otherwise one simple root at $-3a/b; -3a/c$ and two coincident at $-3a/(2b) ; -3a/(2c)$;

*$\Delta < 0$ the equation has only one real root.

Then, denoting by $c_1, c_2, c_3$ the roots of the equation in $\cos$, eventually reduced to only two or one,
and by $s_1, s_2, s_3$ those of the equation in $\sin$, you shall impose to have at least a couple for which
$$
\left| {c_{\,k} } \right| \le 1\quad  \wedge \quad \left| {s_{\,k} } \right| \le 1\quad
  \wedge \quad c_{\,k} ^2  + s_{\,k} ^2  = 1
$$
That means to write each cubic as the product of the three monomials $(x-c_k)$, etc.
and patiently translate the above bounds and conditions onto $a,b,c$ by means of Vieta's formulas.
We get in fact
$$
\left\{ \begin{array}{l}
 c_1  + c_2  + c_3  = 0 \\  c_1 c_2  + c_1 c_3  + c_2 c_3  = a \\  c_1 c_2 c_3  = b \\ 
 \end{array} \right.\quad  \Rightarrow \quad \left\{ \begin{array}{l}
 c_1  =  - \left( {c_2  + c_3 } \right) \\  c_2 ^2  + c_2 c_3  + c_3 ^2 
 = \frac{3}{4}\left( {c_2  + c_3 } \right)^2  + \frac{1}{4}\left( {c_2  - c_3 } \right)^2  =  - a \\ 
 \left( {c_2  + c_3 } \right)c_2 c_3  =  - b \\ 
 \end{array} \right.
$$
and similarly for the sine, changing $b$ with $c$.
So that, if the three solutions are real, the ellipse in the second line demand $a$ to be negative, resulting
in the two curves are sketched below.

Note that, the properties of the cubic ensure that there is a minimum and maximum value
for $b$ (in function of $a$) for which the two curves simultaneously touch each other,
and such values are easy to determine imposing the tangency along $c_2 = c_3$.
And we shall impose that that point at least be within $[-1,1] $, etc.
Finally note that the intersection points are six, since we have not ordered the roots.
