If $$\theta$$ denotes the acute angle between the curves, $$y = 10 - x^2$$ and $$y = 2 + x^2$$ at a point of their intersection, then $$\tan\theta$$ is equal to:

First we locate the points where the two curves intersect. At an intersection point both $$y$$-values are the same, so we write

$$10 - x^2 = 2 + x^2.$$

Now we collect all terms on one side:

$$10 - 2 = x^2 + x^2.$$

$$8 = 2x^2.$$

Dividing by $$2$$ gives

$$x^2 = 4.$$

Taking square roots, we obtain the two abscissae

$$x = 2 \quad \text{or} \quad x = -2.$$

We substitute either value back into either original equation to find the common ordinate. Using $$y = 10 - x^2$$, we get

For $$x = 2$$: $$y = 10 - (2)^2 = 10 - 4 = 6.$$ For $$x = -2$$: $$y = 10 - (-2)^2 = 10 - 4 = 6.$$

Thus the curves meet at the two points $$(2,\,6)$$ and $$(-2,\,6).$$

The acute angle between the curves at an intersection is the acute angle between their tangents. Hence we compute the slopes (derivatives) of each curve.

For $$y = 10 - x^2$$ we differentiate to get

$$\frac{dy}{dx} = -2x.$$

For $$y = 2 + x^2$$ we differentiate to get

$$\frac{dy}{dx} = 2x.$$

We now evaluate these derivatives at the intersection points.

At $$x = 2$$: First curve slope $$m_1 = -2(2) = -4,$$ Second curve slope $$m_2 = 2(2) = 4.$$

At $$x = -2$$: First curve slope $$m_1 = -2(-2) = 4,$$ Second curve slope $$m_2 = 2(-2) = -4.$$

In either case the pair of slopes is $$-4$$ and $$4$$, merely interchanged, so the magnitude of the angle between the tangents is the same.

The formula for the tangent of the angle $$\theta$$ between two lines with slopes $$m_1$$ and $$m_2$$ is

$$\tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|.$$

Substituting $$m_1 = -4$$ and $$m_2 = 4$$, we have

$$\tan\theta = \left|\frac{4 - (-4)}{1 + (-4)(4)}\right|.$$

Simplifying step by step,

$$\tan\theta = \left|\frac{4 + 4}{1 - 16}\right| = \left|\frac{8}{-15}\right| = \frac{8}{15}.$$

Because we have taken the absolute value, the result is positive and already corresponds to the acute angle (i.e. an angle < $$90^\circ$$).

Thus

$$\tan\theta = \frac{8}{15}.$$

Hence, the correct answer is Option D.