<s>[INST] If $y = \cos(2\theta)+\sin(2\theta)$ and $x = \cos(2\theta)-\sin(2\theta)$, express $y$ in terms of $x$. If $y = \cos(2\theta)+\sin(2\theta)$ and $x = \cos(2\theta)-\sin(2\theta)$, express $y$ in terms of $x$.
a) $y=\pm \sqrt{2+x^2}$
b) $y=\pm \sqrt{4-x^2}$
c) $y=\pm \sqrt{2-x^2}$
d) $y=\pm \sqrt{2-2x^2}$
I tried to find relations between the 2 but I'm finding it quite difficult and even after subbing them in, I couldn't find anything. Which of the multiple choice is correct? Please help! Thank you!
 [/INST] Once you observe 
$$
y^2 = 1 + 2 \cos 2 \theta \sin 2 \theta\\ 
x^2 = 1 - 2 \cos 2 \theta \sin 2 \theta
$$
It is clear that  $y^2 + x^2 = 2$. Now all you have to prove is the statements above. Note there is nothing special about $2 \theta$ , any angle there and the same works. This is a generalization of $(x-y)^2 + (x+y)^2  = 2(x^2+y^2)$, and if $x^2+y^2$ can be simplified, this expression can be simplified as well.
</s>