If x + y = 3 - 4θ and x - y = 4 2θ , then the value of x + y is...

Adding both the given equations, we get,

$2x = 3 + 4\sin 2\theta - \cos 4\theta$
= 3 + 4\sin 2\theta - \left(1 - 2\sin^{2} 2\theta\right)<br />= 2\sin^{2} 2$\theta +$4\sin 2$\theta +$2$<br />$⇒ x = \sin^{2} 2$\theta +$2\sin 2$\theta +$1$<br />$⇒ x = {\left($\sin 2$\theta +1\right)}^{2}
Similarly, $2y = 3 - \cos 4$\theta -$4\sin 2$\theta
$2y = 3 - 4\sin 2$\theta - \left($1 - 2\sin^{2}$2\theta\right)
$= 2\sin^{2}$2\theta -$4sin$2\theta +$2$
$⇒ y = \sin^{2}$2\theta -$2sin$2\theta +$1$
⇒ y = \left(sin $2\theta -$1\right)2<br />\sqrt{x} = \left|sin $2\theta +$1\right| = 1 + sin 2\theta<br />\sqrt{y} = \left|sin $2\theta -$1\right| = 1 - sin $2\theta<br />$⇒ $\sqrt{x} + \sqrt{y} =$2$</p>$