# Question #0d91d

Jun 17, 2016

$y = \left(\frac{3}{4}\right) \left(2 - {x}^{2}\right) .$

#### Explanation:

Recall the identity : ${\sin}^{2} \theta = \frac{1 - \cos 2 \theta}{2.}$

Hence,
$y = 3 {\sin}^{2} \theta = \left(\frac{3}{2}\right) \left(1 - \cos 2 \theta\right) .$..............(1)

But, it is given that $x = \sqrt{2 \cos 2 \theta} ,$
so that ${x}^{2} / 2 = \cos 2 \theta .$

Now, putting this value of $\cos 2 \theta$ in (1), we get,

$y = \left(\frac{3}{2}\right) \left(1 - {x}^{2} / 2\right) = \left(\frac{3}{4}\right) \left(2 - {x}^{2}\right) .$

Jun 17, 2016

$y = \frac{{x}^{2} - 2}{-} 2$

#### Explanation:

$y = 3 {\sin}^{2} \theta$
$x = \sqrt{2 \cos 2 \theta}$
${x}^{2} = 2 \cos 2 \theta$
=$2 {\cos}^{2} \theta - 2 {\sin}^{2} \theta$
=$2 {\cos}^{2} \theta - \frac{2}{3} y$
=$2 \left(1 - \frac{1}{3} y\right) - \frac{2}{3} y$
=$2 - \frac{4}{3} y$
so
$y = - \frac{3}{4} \left({x}^{2} - 2\right)$