Trigonometry & Inverse Trigonometry
Logarithmic trigonometric equation; inverse trig
nta_pyq_2023_jan
Grade 12
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:
Step-by-Step Solution
Key Concept: Convert logarithms to natural log form, let $u = \ln\sin x / \ln\cos x$, reduce to a quadratic, solve for $\sin x$.
$\sin x = \frac{-1+\sqrt{5}}{2}$, so $x = \sin^{-1}\left(\frac{-1+\sqrt{5}}{2}\right)$. $\alpha = -1$, $\beta = 5$. $\alpha+\beta = 4$. Answer: (4)
Correct Answer: 4