The smallest positive value of `x` (in radians) satisfying the equation `(log)_(cosx)((sqrt(3))/2sinx)=2-(log)_(secx)(tanx),` is `pi/(12)` (b) `pi/6` (c) `pi/4` (d) `pi/3`

Options

A. `pi/12`
B. `pi/6`
C. `pi/4`
D. `pi/3`

Explanation

To solve the equation \( \log_{\cos x} \left( \frac{\sqrt{3}}{2} \sin x \right) = 2 - \log_{\sec x} (\tan x) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \log_{\cos x} \left( \frac{\sqrt{3}}{2} \sin x \right) = 2 - \log_{\sec x} (\tan x) \]

### Step 2: Change the logarithm base
Recall that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). We can rewrite the right-hand side:
\[ \log_{\sec x} (\tan x) = \log_{\frac{1}{\cos x}} \left( \frac{\sin x}{\cos x} \right) \]
Using the change of base formula, we can express this as:
\[ \log_{\sec x} (\tan x) = -\log_{\cos x} \left( \frac{\sin x}{\cos x} \right) = -\left( \log_{\cos x} (\sin x) - \log_{\cos x} (\cos x) \right) \]
Thus, we can rewrite the equation as:
\[ \log_{\cos x} \left( \frac{\sqrt{3}}{2} \sin x \right) = 2 + \log_{\cos x} (\sin x) - 1 \]
This simplifies to:
\[ \log_{\cos x} \left( \frac{\sqrt{3}}{2} \sin x \right) = 1 + \log_{\cos x} (\sin x) \]

### Step 3: Combine logarithms
Using the property of logarithms \( \log_a b + \log_a c = \log_a (bc) \), we can combine the right-hand side:
\[ \log_{\cos x} \left( \frac{\sqrt{3}}{2} \sin x \right) = \log_{\cos x} (2 \sin x) \]

### Step 4: Set the arguments equal
Since the logarithms are equal, we can set the arguments equal to each other:
\[ \frac{\sqrt{3}}{2} \sin x = 2 \sin x \]

### Step 5: Simplify the equation
Rearranging gives:
\[ \frac{\sqrt{3}}{2} \sin x - 2 \sin x = 0 \]
Factoring out \( \sin x \):
\[ \sin x \left( \frac{\sqrt{3}}{2} - 2 \right) = 0 \]

### Step 6: Solve for \( x \)
This gives us two cases:
1. \( \sin x = 0 \) which leads to \( x = n\pi \) (not positive).
2. \( \frac{\sqrt{3}}{2} - 2 = 0 \) which has no solution since \( \frac{\sqrt{3}}{2} < 2 \).

### Step 7: Solve \( \cos x = \frac{\sqrt{3}}{2} \)
From the equation \( \cos x = \frac{\sqrt{3}}{2} \):
\[ x = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x = -\frac{\pi}{6} + 2n\pi \]
The smallest positive solution is:
\[ x = \frac{\pi}{6} \]

### Conclusion
Thus, the smallest positive value of \( x \) satisfying the equation is:
\[ \boxed{\frac{\pi}{6}} \]