All the pairs (x, y) that satisfy the inequal | vedclass

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Question

$2^{\sqrt{\sin ^{2} x-2 \sin x+5}}-4^{-\sin ^{2} y} \leq 1$

$ \Rightarrow \,{2^{\sqrt {{{(\sin \,x - 1)}^2} + 4} }}\, \leqslant {4^{{{\sin }^2}y}}$

Vedclass · Accepted Answer

$\sin \,x = \left| {\sin \,y} \right|$

Vedclass · Answer

$2^{\sqrt{\sin ^{2} x-2 \sin x+5}}-4^{-\sin ^{2} y} \leq 1$

$ \Rightarrow \,{2^{\sqrt {{{(\sin \,x - 1)}^2} + 4} }}\, \leqslant {4^{{{\sin }^2}y}}$

$\sqrt {\frac{{\underbrace {{{(\sin \,x - 1)}^2}}_{ \geqslant 0} + 4}}{{ \geqslant 2}}}  \leqslant \underbrace {2\,{{\sin }^2}\,y}_{ \leqslant 2}$

this is possible only if $\sin x=1$ and $|\sin y|=1$

All the pairs $(x, y)$ that satisfy the inequality ${2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1$ also Satisfy the equation

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