Question

If $x+y=3-cos4\theta$ and $x-y=4sin2\theta$ , then the value of $\sqrt{x}+\sqrt{y}$ is equal to

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (A)

Adding both the given equations, we get,
$2x=3+4sin2\theta -cos4\theta$
$=3+4sin2\theta -\left(1-2{\left(sin\right)}^{2}2\theta \right)$
$=2si{n}^{2}2\theta +4sin2\theta +2$
$\Rightarrow x=si{n}^{2}2\theta +2sin2\theta +1$
$\Rightarrow x={\left(sin2\theta +1\right)}^2$
Similarly, $2y=3-cos4\theta -4sin2\theta$
$2y=3-4sin2\theta -\left(1-2{\left(sin\right)}^{2}2\theta \right)$
$=2si{n}^{2}2\theta -4sin2\theta +2$
$\Rightarrow y=si{n}^{2}2\theta -2sin2\theta +1$
$\Rightarrow y={\left(sin2\theta -1\right)}^2$
$\sqrt{x}=\left|sin2\theta +1\right|=1+sin2\theta$
$\sqrt{y}=\left|sin2\theta -1\right|=1-sin2\theta$
$\Rightarrow \sqrt{x}+\sqrt{y}=2$