$x \cos \theta – y \sin \theta = \cos 2\theta$
$x \sin \theta + y \cos \theta = 2 \sin 2\theta$
I tried to use cross multiplication method to find $\cos \theta$ and $\sin \theta$ and then put the values in $\cos^2 \theta + \sin^2 \theta = 1$, but was not able to eliminate $\cos 2\theta$ or $\sin 2\theta$. Please help me in solving this question.
Best Answer
Multplying the first equation by $\sin \theta$ gives
$$x \cos\theta\sin \theta -y\sin^2\theta = \sin\theta \cos 2\theta\tag{1}$$
and the second by $\cos \theta$ will give
$$ x \cos\theta\sin \theta + y \cos^2\theta = 2\sin 2\theta \cos \theta\tag{2}$$
so we can eliminate the $x$ by subtracting 1 from 2:
$$y (\cos^2 \theta + \sin^2 \theta) = 2\sin 2\theta \cos \theta - \sin\theta \cos 2\theta$$
so $$y = 2\sin 2\theta \cos \theta - \sin\theta \cos 2\theta$$
Now we can maybe simplify by using the double angle formulae:
$$\sin 2\theta = 2\sin \theta \cos \theta$$ and
$$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$$
Try it. And then solve $x$ by a similar multiplication and substraction or adding.