If the solution of the equation \(\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1,\; x \in \left(0,\dfrac{\pi}{2}\right),\) is \(\sin^{-1}\!\left(\dfrac{\alpha + \sqrt{\beta}}{2}\right),\) where \(\alpha, \beta\) are integers, then \(\alpha + \beta\) is equal to:
