<s>[INST] How do you simplify an expression involving fourth and higher order trigonometric functions? The problem is as follows:
Which value of $K$ has to be in order that $R$ becomes independent from $\alpha$?.
$$R=\sin^6\alpha +\cos^6\alpha +K(\sin^4\alpha +\cos^4\alpha )$$
So far I've only come up with the idea that the solution may involve $R=0$, therefore
$$\sin^6\alpha +\cos^6\alpha +K(\sin^4\alpha +\cos^4\alpha)=0$$ 
as a result the expression becomes $0$ thus independent from $\alpha$, however the result is like this
$$-K=\frac{\sin^6\alpha +\cos^6\alpha}{\sin^4\alpha +\cos^4\alpha}$$
I am not sure if this is the right way.
Moreover, how can I simplify this expression, as it has order four and six?
 [/INST] $$1=(\sin^2x+\cos^2x)^3= \sin^6x+ \cos^6x+ 3\sin^2x\cos^2x(\sin^2x+\cos^2x)= \sin^6x+ \cos^6x+ 3\sin^2x\cos^2x$$
$$3\sin^2x\cos^2x=\frac{3}{2}((\sin^2x+\cos^2x)^2 - \sin^4x-\cos^4x)=\frac{3}{2}((1- \sin^4x-\cos^4x))$$
Substitute and compare with the equation you have
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