If ` cos x = (12)/( 13)` and x is an acute angle, then
`sqrt((1 + (sin x)/(cos x))( 1- tan x)) ` is

Options

A. `(sqrt(115))/(12)`
B. ` (sqrt(116))/( 12)`
C. ` (sqrt(119))/( 12)`
D. ` (sqrt(117))/( 12)`

Answer

To solve the problem, we start with the given information:

**Given:**
\[ \cos x = \frac{12}{13} \]
and \(x\) is an acute angle.

We need to evaluate the expression:
\[ \sqrt{\left(1 + \frac{\sin x}{\cos x}\right)\left(1 - \tan x\right)} \]

### Step 1: Find \(\sin x\)
Using the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
we can find \(\sin x\):
\[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{12}{13}\right)^2 \]
Calculating \(\cos^2 x\):
\[ \cos^2 x = \frac{144}{169} \]
Thus,
\[ \sin^2 x = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]
Taking the square root (since \(x\) is acute):
\[ \sin x = \frac{5}{13} \]

### Step 2: Find \(\tan x\)
Using the definition of tangent:
\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} \]

### Step 3: Substitute values into the expression
Now substitute \(\sin x\) and \(\tan x\) into the expression:
\[ \sqrt{\left(1 + \frac{\sin x}{\cos x}\right)\left(1 - \tan x\right)} = \sqrt{\left(1 + \frac{5/13}{12/13}\right)\left(1 - \frac{5}{12}\right)} \]

### Step 4: Simplify the first term
Calculating \(1 + \frac{5/13}{12/13}\):
\[ 1 + \frac{5}{12} = \frac{12}{12} + \frac{5}{12} = \frac{17}{12} \]

### Step 5: Simplify the second term
Calculating \(1 - \tan x\):
\[ 1 - \frac{5}{12} = \frac{12}{12} - \frac{5}{12} = \frac{7}{12} \]

### Step 6: Combine the two terms
Now we substitute back into the expression:
\[ \sqrt{\left(\frac{17}{12}\right)\left(\frac{7}{12}\right)} = \sqrt{\frac{119}{144}} \]

### Step 7: Final simplification
Taking the square root:
\[ \sqrt{\frac{119}{144}} = \frac{\sqrt{119}}{12} \]

### Conclusion
Thus, the final answer is:
\[ \frac{\sqrt{119}}{12} \]

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