$\large \left( \log_{\cos x} \sin^2 x \right) \left( \log_{\cos^2 x} \sin x \right) = 4$

If $x = \arcsin\left( \dfrac{\sqrt {\color{red}{y}} - 1}2 \right) + 2k \pi$ for any integer $k$, find $\color{red}{y}$.

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