JEE Mains · Maths · STD 11 - Trigonometrical equations
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 \(4\)
Answer & Solution
Correct Answer
(D) \(4\)
Step-by-step Solution
Detailed explanation
\(\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1\) \(\Rightarrow \frac{\ln \cos x-\ln \sin x}{\ln \cos x}+4 \frac{\ln \sin x-\ln \cos x}{\ln \sin x}=1\) \(\Rightarrow(\ln \sin x)^2-4(\ln \sin x)(\ln \cos x)+4(\ln \cos x)^2=1\) \(\Rightarrow \ln \sin x=2 \ln \cos x\)…
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