Simplifying a trigonometric equation Given $\alpha\in\mathbb{R}, \ \alpha\neq1 $  and $\beta\in\mathbb{R}, \ $ find the values of $ \ \theta \ $ such that 
$$ \alpha \cos\left(\beta \theta \right)=\cos\left(\left(\beta+1\right) \theta \right)\,$$
in terms of $\alpha \ $ and $ \ \beta$.
 A: This is a comment, not an answer, but I find it interesting to generalise this question - WLOG - to:
Given $a\in\mathbb{R}, \ b\in\mathbb{R}, \ \alpha\in\mathbb{R}, \ \alpha\neq1 , \ $ $\beta\in\mathbb{R}, \ $ and $ \ \gamma\in\mathbb{R} , \ $ find the values of $ \ \theta \ $ such that 
$$ \sin \theta = a \ \sin\left(\alpha \theta +  \beta \right)\ + b \ ,$$
in terms of $\alpha \ $ and $ \ \beta$.
e.g. $ \sin x = 5 \sin (2x-6) + 3 \ $ looks like:
https://www.wolframalpha.com/input/?i=sinx+%3D+5sin%282x-6%29+%2B+3
although you don't need to use integers.
This basically is asking: "find all the intersections of any two sinusoidal waves". 
But why stop there? You could also think even more generally and think, "what are the solutions to a line intersecting a sinusoidal wave, e.g. https://www.wolframalpha.com/input/?i=x%2F20+%3D+sin%28x%2B1%29
Again, I suspect numerical methods are in order.
Or a quadratic and a sine wave? The possibilities are endless...
I'll probably post these as questions on SE at some point.
