Prove the following


All the pairs $(\mathrm{x}, \mathrm{y})$ 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 equation:

  1. (1) $2|\sin x|=3 \sin y$

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

  3. (3) $\sin x=2 \sin y$

  4. (4) $\sin x=|\sin y|$

Correct Option: , 4


Given inequality is,

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

$\Rightarrow \sqrt{\sin ^{2} x-2 \sin x+5} \leq 2 \sin ^{2} y$

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

It is true if $\sin x=1$ and $|\sin y|=1$

Therefore, $\sin x=|\sin y|$

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