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)

4

Answer 2

The correct answer is (B) : 4
logcosx​cotx+4logsinx​tanx=1
⇒lncosxlncosx−lnsinx​+4lnsinxlnsinx−lncosx​=1
⇒(lnsinx)2−4(lnsinx)(lncosx)+4(lncosx)2=1
⇒lnsinx=2lncosx
⇒sin2x+sinx−1=0⇒sinx=2−1+5​​
∴α+β=4