The value of x, satisfying the equation `log_(cos x ) sin x =1`, where `0 lt x lt (pi)/(2)`, is

Options

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

Answer

To solve the equation \( \log_{\cos x} \sin x = 1 \) for \( 0 < x < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the logarithmic equation
The equation \( \log_{\cos x} \sin x = 1 \) can be rewritten in exponential form:
\[ \sin x = \cos x^1 \]
This means:
\[ \sin x = \cos x \]

### Step 2: Use the identity of sine and cosine
From the equation \( \sin x = \cos x \), we can divide both sides by \( \cos x \) (since \( \cos x \neq 0 \) in the interval \( 0 < x < \frac{\pi}{2} \)):
\[ \tan x = 1 \]

### Step 3: Solve for x
The equation \( \tan x = 1 \) implies that:
\[ x = \frac{\pi}{4} \]
This is the only solution in the interval \( 0 < x < \frac{\pi}{2} \).

### Conclusion
Thus, the value of \( x \) satisfying the equation \( \log_{\cos x} \sin x = 1 \) is:
\[ \boxed{\frac{\pi}{4}} \]