The domain of the function `f(x)=(tan 2x)/(6 cos x+2 sin x2x)" is "`

Options

A. `R-{(2n+1)pi/2: n in z}`
B. `R-{(2n+1)pi/4: n in z}`
C. `R-{{(2n+1)pi/2: n in Z} cup {(2n+1) pi/4: n in z}}`
D. None of these

Answer

To find the domain of the function \( f(x) = \frac{\tan(2x)}{6 \cos x + 2 \sin(2x)} \), we need to ensure that the function is defined. This means we need to identify the values of \( x \) for which the denominator is not equal to zero and where the tangent function is defined.

### Step 1: Identify when \( \tan(2x) \) is defined
The tangent function is undefined when its argument is an odd multiple of \( \frac{\pi}{2} \). For \( \tan(2x) \), we set:
\[ 2x = \frac{(2n + 1)\pi}{2} \quad \text{for } n \in \mathbb{Z} \]
This simplifies to:
\[ x = \frac{(2n + 1)\pi}{4} \]
Thus, \( x \) cannot take these values.