QJEEM23Mabf86: JEE Main 2023 Mathematics - Class 12 JEEM23Mabf86 Solution: If the solution of the equation , is , where , are integers,…

${\log _{\cos x}}\cot x + 4{\log _{\sin x}}\tan x = 1$

$\Rightarrow {\log _{\cos x}}\cot x - 4{\log _{\sin x}}\cot x = 1$

$\Rightarrow 1 - {\log _{\cos x}}\sin x - 4 - 4{\log _{\sin x}}\cos x = 1$

Let ${\log _{\cos x}}\sin x = t$

$t + {4 \over t} = 4$

$\Rightarrow t = 2$

$\sin x = {\cos ^2}x$

$\Rightarrow \sin x = 1 - {\sin ^2}x$

$\Rightarrow {\sin ^2}x + \sin {x^{ - 1}} = 0$

$\Rightarrow \sin x = {{ - 1\, \pm \,\sqrt 5 } \over 2}$

as $x \in \left( {0,{\pi \over 2}} \right)$

$\sin x = {{\sqrt 5 - 1} \over 2}$

$x = {\sin ^{ - 1}}\left( {{{ - 1 + \sqrt 5 } \over 2}} \right)$

$\Rightarrow \alpha = - 1,\beta = 5$

$\alpha + \beta = 4$