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 = |siny|
• B. sin x = 2 sin y
• C. 2|sinx| = 3siny
• D. 2sin x = siny

Answer 1

Answer: (A)

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