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Step-by-Step Solution
Step 1: Rewrite the given inequality in terms of powers of 2
We have the inequality
$2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot \dfrac{1}{4^{\sin^2 y}} \le 1.$
Recall that $4^{\sin^2 y} = (2^2)^{\sin^2 y} = 2^{2\sin^2 y}.$ Therefore,
$\dfrac{1}{4^{\sin^2 y}} = 2^{-2\sin^2 y}.$
Hence, the inequality becomes
$
2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot 2^{-2\sin^2 y}
\;\le\; 1.
$
Step 2: Combine the exponents of 2
Combine the two powers of 2:
\[
2^{\sqrt{\sin^2 x - 2\sin x + 5} \;-\; 2\sin^2 y} \;\le\; 1.
\]
A power of 2 is less than or equal to 1 precisely when its exponent is less than or equal to 0, so we set up the inequality:
\[
\sqrt{\sin^2 x - 2\sin x + 5} \;-\; 2\sin^2 y \;\le\; 0.
\]
This implies
\[
\sqrt{\sin^2 x - 2\sin x + 5} \;\le\; 2\sin^2 y.
\]
Step 3: Simplify the expression under the square root
Note that
\[
\sin^2 x - 2\sin x + 5
= (\sin x - 1)^2 + 4.
\]
Thus the inequality becomes
\[
\sqrt{(\sin x - 1)^2 + 4} \;\le\; 2\sin^2 y.
\]
Step 4: Analyze the minimum and maximum values
1. The left side, $\sqrt{(\sin x - 1)^2 + 4}$, is always greater than or equal to $2$, because $(\sin x - 1)^2 \ge 0$ implies $(\sin x - 1)^2 + 4 \ge 4$ and taking the square root gives us a minimum possible value of $2.$
2. The right side, $2\sin^2 y,$ has a maximum possible value of $2$ (since $\sin^2 y \le 1$).
For the inequality
\[
\sqrt{(\sin x - 1)^2 + 4} \;\le\; 2\sin^2 y
\]
to hold, both sides must simultaneously reach these extremal values:
\[
\sqrt{(\sin x - 1)^2 + 4} = 2
\quad\text{and}\quad
2\sin^2 y = 2.
\]
Step 5: Solve for sin x and sin y
• From $\sqrt{(\sin x - 1)^2 + 4} = 2,$ we get
\[
(\sin x - 1)^2 + 4 = 4
\;\Rightarrow\; (\sin x - 1)^2 = 0
\;\Rightarrow\; \sin x = 1.
\]
• From $2\sin^2 y = 2,$ we get
\[
\sin^2 y = 1
\;\Rightarrow\; \sin y = \pm 1
\;\Rightarrow\; |\sin y| = 1.
\]
Putting these together, when $\sin x = 1,$ it matches $|\sin y|=1.$ Indeed, the relationship
\[
\sin x = |\sin y|
\]
is satisfied.
Conclusion
Thus, all pairs $(x, y)$ satisfying the original inequality also satisfy
\[
\sin x = \bigl|\sin y\bigr|.
\]
Therefore, the correct answer is
$\sin x = \vert \sin y \vert.$
