Trigonometric Ratios Question 4
Question 4 - 30 January - Shift 1
If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in(0, \frac{\pi}{2})$, $\sin ^{-1}(\frac{\alpha+\sqrt{\beta}}{2})$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to:
(1) 3
(2) 5
(3) 6
(4) 4
Show Answer
Answer: (4)
Solution:
Formula: Properties of Logarithmic Functions, Roots of equations, Pythagorean Identities
$\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1$
$\Rightarrow \frac{\ln \cos x-\ln \sin x}{\ln \cos x}+4 \frac{\ln \sin x-\ln \cos x}{\ln \sin x}=1$
$\Rightarrow(\ln \sin x)^{2}-4(\ln \sin x)(\ln \cos x)+4(\ln \cos x)^{2}= 1$
$\Rightarrow \ln \sin x=2 \ln \cos x$
$\Rightarrow \sin ^{2} x+\sin x-1=0 \Rightarrow \sin x=\frac{-1+\sqrt{5}}{2}$
$\therefore \alpha+\beta=4$
Correct option (4)
