<s>[INST] If $\sin x - \cos x = \frac{1}{2}$ then determine: $\sin^4 x + \cos^4 x$ If $$\sin x - \cos x = \frac{1}{2}$$ then determine: $$\sin^4 x + \cos^4 x$$
I tried making it $(\sin^2 x)^2+(\cos^2 x)^2$ but then I get nothing that can help. What is the trick to this?
 [/INST] You have
$$
\frac{1}{4}=(\sin x-\cos x)^2=\sin^2x-2\sin x\cos x+\cos^2x
$$
so
$$
\sin x\cos x=\frac{1}{2}\left(1-\frac{1}{4}\right)=\frac{3}{8}
$$
Then
$$
\sin^4x+\cos^4x=(\sin^2x+\cos^2x)^2-\dotsb
$$
</s>