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 :

Question

If $ y = \ln(x^2 + 1) $, then find $ \frac{dy{dx} $ at $ x = 1 $.

Answer 1

The given equation is:

$\log _{\cos x} \cot x + 4 \log _{\sin x} \tan x = 1$

We need to find the solution in the form $\sin^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha$ and $\beta$ are integers, and determine $\alpha + \beta$.

Let's solve the equation step by step:

First, simplify the equation using properties of logarithms:

$\log _{\cos x} \cot x = \frac{\log(\cot x)}{\log(\cos x)}$

And,

$4 \log _{\sin x} \tan x = 4 \frac{\log(\tan x)}{\log(\sin x)}$

The equation thus transforms into:

$\frac{\log(\cot x)}{\log(\cos x)} + 4\frac{\log(\tan x)}{\log(\sin x)} = 1$

We know the following identities:

$\cot x = \frac{\cos x}{\sin x}$ and $\tan x = \frac{\sin x}{\cos x}$

Using the identities,

$\log(\cot x) = \log(\cos x) - \log(\sin x)$

$\log(\tan x) = \log(\sin x) - \log(\cos x)$

Substitute these back into the equation:

$\frac{\log(\cos x) - \log(\sin x)}{\log(\cos x)} + 4 \frac{\log(\sin x) - \log(\cos x)}{\log(\sin x)} = 1$

This simplifies to:

$1 - \frac{\log(\sin x)}{\log(\cos x)} + 4 \left(1 - \frac{\log(\cos x)}{\log(\sin x)}\right) = 1$

Simplify the expression further:

$1 - \frac{\log(\sin x)}{\log(\cos x)} + 4 - 4 \frac{\log(\cos x)}{\log(\sin x)} = 1$

$5 - \frac{\log(\sin x)}{\log(\cos x)} - 4 \frac{\log(\cos x)}{\log(\sin x)} = 1$

$4 = \frac{\log(\sin x)}{\log(\cos x)} + 4 \frac{\log(\cos x)}{\log(\sin x)}$

Upon rearranging terms and solving the inequality, it eventually leads to:

$\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1$

Therefore, simplifying gives us that $x = \sin^{-1}(\pm \frac{\sqrt{3}}{2})$, which is valid between $\left(0, \frac{\pi}{2}\right)$.

The recognized solution form is:

$\sin^{-1}\left(\frac{1 + \sqrt{3}}{2}\right)$

Thus, $\alpha = 1$ and $\beta = 3$.

Summing them gives:

$\alpha + \beta = 1 + 3 = 4$

The correct answer is therefore: 4.

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 :

The Correct Option is B

Solution and Explanation

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