Solving trigonometric equations like $1-s\sin^2\theta=a\sin^6\theta+b\cos^6\theta$

Question

For what values of $s\in\mathbb{R}$ does the following identity hold for all $\theta\in\mathbb{R}$:

$$1-s\cos^2\theta\sin^2\theta=a\sin^6\theta+b\cos^6\theta\tag{1}$$

for some $a,b\in\mathbb{R}$? In other words, for what values of $s$ can the expression $1-s\cos^2\theta\sin^2\theta$ be simplified to $a\sin^6\theta+b\cos^6\theta$ for some $a$ and $b$ not depending on $s$ or $\theta$? I know that $s=3$ holds, since we have the identity:

$$1-3\cos^2\theta\sin^2\theta=\sin^6\theta+\cos^6\theta$$

but I would like to know (purely out of interest) if there are other values of $s$ for which it holds. I would also like to know for what if any values of s the following identity holds:

$$1-s\cos^2\theta=a\sin^6\theta+b\cos^6\theta\tag{2}$$

for some $a,b\in\mathbb{R}$; and I would also like to know for what if any values of $s$ we have the following identity:

$$1-s\sin^2\theta=a\sin^6\theta+b\cos^6\theta\tag{3}$$

I am not aware of any theory that would help me solve trigonometric equations like $(1)$, $(2)$ and $(3)$ except by trial and error, but I would love to know if there is a method. I assume that there are only particular values of $s$ for which equations like these can hold, but I can not see any way of finding them.

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The context of this question relates to a question I recently asked here, where I was attempting to evaluate integrals of the form $\int_0^\infty\left(a+\sin^2{\theta}\right)^{-\frac{1}{3}}\;d\theta$; there are complicated reasons as to which values of $a$ give nice values, but I noticed that the simple method I was using could possibly be extended for certain values of $a$ if I knew the solutions to equations $(1)$, $(2)$ and $(3)$ above. Thus I am wondering if anybody knows how to solve these equations

Answer 1

Since the equality should hold for all $\theta$, try first with $\theta=0$, that gives $b=1$. Then for $\theta=\pi/2$ we get $a=1$.

Let's try now with $\theta=\pi/4$: $$ 1-s\frac{1}{2}\frac{1}{2}=\frac{1}{8}+\frac{1}{8} $$ so $s=3$.

It should be easy to apply a similar method for showing the other two are not identities for any value of $s,a,b$.

The identity $1-3\cos^2\theta\sin^2\theta=\sin^6\theta+\cos^6\theta$ is indeed true, but the argument above doesn't prove it: we only got necessary conditions. However, it's easy to derive it as a consequence of $$ x^3+y^3=(x+y)^3-3xy(x+y) $$ with $x=\sin^2\theta$ and $y=\cos^2\theta$.