Question

All the pairs $(x,y)$ that satisfy the inequality $2\sqrt{{\sin }^{2}x-2\sin x+5}.\frac{1}{4^{{\sin }^{2}y}}\le 1$ also satisfy the eauation.

• A. $\sin x=|\sin y|$
• B. $\sin x=2\sin y$
• C. $2|\sin x|=3\sin y$
• D. $2\sin x=\sin y$

Answer 1

Answer: (A)

$\sin x=|\sin y|$

$2\sqrt{{\sin }^{2}x-2\sin x+5}\cdot 4^{-{\sin }^{2}y}\le 1$
$\Rightarrow 2\sqrt{(\sin x-1{)}^2+4}\le 2^{2{\sin }^{2}y}$
$\Rightarrow \sqrt{(\sin x-1{)}^2+4}\le 2{\sin }^{2}y$
$\Rightarrow \sin x=1$ and $|\sin y|=1$