The correct option is 13/3.

Based on the provided image, we are given the following trigonometric ratio:
$$\sin A=\frac{4}{5}$$
We need to evaluate the expression:
$$(3-\tan A)(2+\cos A)$$

To find the values of $\cos A$ and $\tan A$, we use the definition of the sine function in a right-angled triangle:
$$\sin A=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{4}{5}$$
Let us consider a right-angled triangle where:
Opposite side = 4
Hypotenuse = 5

Using Pythagoras' theorem, we can determine the length of the adjacent side:
$${\text{Adjacent}}^2+{\text{Opposite}}^2={\text{Hypotenuse}}^2$$
$${\text{Adjacent}}^2+4^2=5^2$$
$${\text{Adjacent}}^2+16=25$$
$${\text{Adjacent}}^2=25-16$$
$${\text{Adjacent}}^2=9$$
$$\text{Adjacent}=\sqrt{9}=3$$

Now, we can find the values of $\cos A$ and $\tan A$:
$$\cos A=\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{3}{5}$$
$$\tan A=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{4}{3}$$

Next, substitute these values into the given expression:
$$(3-\tan A)(2+\cos A)=(3-\frac{4}{3})(2+\frac{3}{5})$$

Simplify each factor separately:
First factor:
$$3-\frac{4}{3}=\frac{9-4}{3}=\frac{5}{3}$$
Second factor:
$$2+\frac{3}{5}=\frac{10+3}{5}=\frac{13}{5}$$

Multiply the two simplified results together:
$$\left(\frac{5}{3}\right)\times \left(\frac{13}{5}\right)=\frac{5\times 13}{3\times 5}$$
Canceling the common factor of 5 in the numerator and denominator gives:
$$\frac{13}{3}$$