Question

If $x+y=3-cos 4 \theta $ and $x-y=4sin 2 \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+4sin 2 \theta - cos ⁡ 4 \theta $
$=3+4sin 2 \theta - \left(1 - 2 \left(sin\right)^{2} ⁡ 2 \theta \right)$
$=2sin^{2} 2 \theta + 4 sin ⁡ 2 \theta + 2$
$\Rightarrow x=sin^{2} 2 \theta + 2 sin ⁡ 2 \theta + 1$
$\Rightarrow x=\left(sin 2 \theta + 1\right)^{2}$
Similarly, $2y=3-cos 4 \theta - 4 sin ⁡ 2 \theta $
$2y=3-4sin 2 \theta - \left(1 - 2 \left(sin\right)^{2} ⁡ 2 \theta \right)$
$=2sin^{2} 2 \theta - 4 sin ⁡ 2 \theta + 2$
$\Rightarrow y=sin^{2} 2\theta -2sin⁡2\theta +1$
$\Rightarrow y=\left(sin 2 \theta - 1\right)^{2}$
$\sqrt{x}=\left|sin 2 \theta + 1\right|=1+sin ⁡ 2 \theta $
$\sqrt{y}=\left|sin 2 \theta - 1\right|=1-sin ⁡ 2 \theta $
$\Rightarrow \sqrt{x}+\sqrt{y}=2$