If the solution of the equation \( \log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1 \), \( x \in\left(0, \frac{\pi}{2}\ri…

If the solution of the equation

\( \log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1 \),

\( x \in\left(0, \frac{\pi}{2}\right) \), is \( \sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right) \), where \( \alpha \beta \) are

integers, then \( \alpha+\beta \) is equal to:

[JEE Main 2023, 30 Jan (Shift 1)]