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

Answer: (B)

$ \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$
Correct option (4)