All the pairs (x, y) that satisfy the inequality

Question:

All the pairs (x, y) that satisfy the inequality

$2 \sqrt{\sin ^{2} x-2 \sin x+5} \cdot \frac{1}{4^{\sin ^{2} y}} \leq 1$ also satisfy the eauation

  1. $\sin x=|\sin y|$

  2. $\sin x=2 \sin y$

  3. $2|\sin x|=3 \sin y$

  4. $2 \sin x=\sin y$

Correct Option: 1

Solution:

$2^{\sqrt{\sin ^{2} x-2 \sin x+5}} \cdot 4^{-\sin ^{2} y} \leq 1$

$\Rightarrow 2^{\sqrt{(\sin x-1)^{2}+4}} \leq 2^{2 \sin ^{2} y}$

$\Rightarrow \sqrt{(\sin x-1)^{2}+4} \leq 2 \sin ^{2} y$

$\Rightarrow \sin x=1$ and $|\sin y|=1$

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