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Trigonometrical Equations
hard
If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right), \quad$ is $\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to:
A
$3$
B
$5$
C
$6$
D
$4$
(JEE MAIN-2023)
Solution
$\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1$
$\Rightarrow \frac{\ln \cos x-\ln \sin x}{\ln \cos x}+4 \frac{\ln \sin x-\ln \cos x}{\ln \sin x}=1$
$\Rightarrow(\ln \sin x)^2-4(\ln \sin x)(\ln \cos x)+4(\ln \cos x)^2=1$
$\Rightarrow \ln \sin x=2 \ln \cos x$
$\Rightarrow \sin ^2 x+\sin x-1=0 \Rightarrow \sin x=\frac{-1+\sqrt{5}}{2}$
$\therefore \alpha+\beta=4$
Standard 11
Mathematics
