The equation of a tangent to the parabola, $$x^2 = 8y$$, which makes an angle $$\theta$$ with the positive direction of x-axis, is

We begin with the given parabola $$x^{2}=8y$$.

First we compare this with the standard form $$x^{2}=4ay$$. By comparison we have $$4a=8\;\Longrightarrow\;a=2.$$

Now we recall the standard result: for the parabola $$x^{2}=4ay,$$ the equation of the tangent having slope $$m$$ is given by the formula

$$y=mx-am^{2}.$$

In the present problem the tangent is said to make an angle $$\theta$$ with the positive direction of the x-axis. The slope of such a line is therefore

$$m=\tan\theta.$$

Substituting $$m=\tan\theta$$ and $$a=2$$ into the tangent formula, we get

$$y=\bigl(\tan\theta\bigr)x-2\bigl(\tan\theta\bigr)^{2}.$$

To compare this with the options, we now isolate $$x$$. Transposing terms yields

$$x\tan\theta=y+2\tan^{2}\theta.$$

Dividing every term by $$\tan\theta$$ (which is non-zero for a valid angle of inclination), we have

$$x=y\cot\theta+2\tan\theta.$$

This equation exactly matches Option C.

Hence, the correct answer is Option C.