Question

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 :

• A. 6
• B. 4
• C. 5
• D. 3

Answer 1

The Correct Option is B

The correct answer is (B) : 4
${\log }_{\cos x}\cot x+4{\log }_{\sin x}\tan x=1$
$\Rightarrow \frac{\ln \cos x-\ln \sin x}{\ln \cos x}+4\frac{\ln \sin x-\ln \cos x}{\ln \sin x}=1$
$\Rightarrow (\ln \sin x{)}^2-4(\ln \sin x)(\ln \cos x)+4(\ln \cos x{)}^2=1$
$\Rightarrow \ln \sin x=2\ln \cos x$
$\Rightarrow {\sin }^{2}x+\sin x-1=0\Rightarrow \sin x=\frac{-1+\sqrt{5}}{2}$
$\therefore \alpha +\beta =4$