On adding and subtracting

$x = \frac{{3 – \cos 4\theta \, + 4\sin 2\theta }}{2}\,$; $y = \frac{{3 – \cos 4\theta \, – 4\sin 2\theta }}{2}\,$

$x =\frac{{4(1 + \sin 2\theta )\, – \,(1 + \cos 4\theta )}}{2}\,$ ; $y =\frac{{4(1 – \sin 2\theta )\, – \,(1 + \cos 4\theta )}}{2}\,$

$x = 2 (1 + sin2\theta ) – cos^22\theta$ ; $y = 2 (1 – sin2\theta ) – cos^22\theta$

$x = 1 + 2\, sin2\theta + sin^22\theta$ ;$ y = 1 – 2 sin2\theta + sin^22\theta$

$x = (1 + sin2\theta )^2$ ;$y = (1 – sin2\theta )^2$ $\Rightarrow \sqrt x \, + \,\sqrt y \, = \,2$

Alternate : Or put $\theta$ = $\frac{\pi }{4}\,$ and verify