If $y = \cos(2\theta)+\sin(2\theta)$ and $x = \cos(2\theta)-\sin(2\theta)$, express $y$ in terms of $x$.

Question

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!

Answer 1

By squaring the given expressions for $x$ and $y$, using the identity $(a+b)^2 = a^2+b^2+2ab$ with appropriate $a,b$, and using the identity $\cos^2(2 \theta) + \sin^2(2 \theta) = 1$, we have $$ y^2 = 1 + 2 \cos 2 \theta \sin 2 \theta\\ x^2 = 1 - 2 \cos 2 \theta \sin 2 \theta $$

It is clear that $y^2 + x^2 = 2$. Hence, $y = \pm \sqrt{2-x^2}$, proving that (c) is the correct option.

Note there is nothing special about the angle $2 \theta$ in this computation. In other words, for any angle $\phi$, if $y= \cos(\phi)+\sin(\phi)$ and $x = \cos(\phi) - \sin(\phi)$, we still have $y = \pm \sqrt{2-x^2}$.

Answer 2

Hint:

Solve the simultaneous equations for $\cos2\theta,\sin2\theta$

Use $$\cos^22\theta+\sin^22\theta=1$$ to eliminate $\theta$